1. Old Chemistry 335, University, MS 38677
ENGR 691 – 73: Introduction to Free-Surface Hydraulics in Open Channels
Lecture 04: Nonuniform Flow
Yan Ding, Ph.D.
Research Assistant Professor, National Center for Computational Hydroscience and Engineering (NCCHE), The University of Mississippi,
Old Chemistry 335, University, MS 38677
Phone: ;
Course Notes by: Mustafa S. Altinakar and Yan Ding

2. Outline Transition Between Subcritical and Critical Flow
Introduction to Hydraulic Jump
Gradually Varied Flow (Governing Equations)
Forms of water surface (Channels on Mild Slope, Critical Slope, Steep Slope, Adverse Slope, Horizontal Slope)
Control Points
Computation of Water Surface (Method of successive Approximations; Method of Direct Integration; Method of Graphical Integration)
Rapidly Varied Flow (Weirs; Spillways; Hydraulic Drop; Underflow Gates; Hydraulic Jump)
Transitions (Channel with variable Bed Floor; Channel of variable Width; Oblique Jump)
Lateral Inflow

3. Transitions between subcritical and critical flow
from subcritical to supercritical flow
Transition
from supercritical to subcritical flow
When the flow changes from subcritical to supercritical the water surface lowers gradually from a higher depth to a lower depth by passing through critical depth.
In the region where the flow changes from subcritical to critical flow, a gradually varied flow takes place.
When the flow changes from supercritical to subcritical the water surface rapidly increases from a supercritical depth to subcritical depth. This sudden increase is called a rapidly varied flow.
The rapidly varied flow may be preceded by a gradually varied flow region where the flow depth rises but stays below critical depth.

4. Introduction to Hydraulic Jump
conjugate depths or
sequent depths
alternate depths
Specific Momentum
Specific Energy
Equation of continuity
Things to remember:
Conjugate depths or sequent depths (on Specific Momentum Curve)
Alternate depths (on Specific Energy Curve)
Momentum equation
with

5. Introduction to Hydraulic Jump
By combining momentum equation and continuity equation, on gets: or
where
and

6. Gradually Varied Flow Equation
Consider the steady non uniform flow in a channel. We wish to develop an equation for the variation of the water surface h(x), i.e. longitudinal water surface profile.
For this, we will consider the equation of energy:
and the equation of continuity:
ref. line
Differentiate the energy equation with respect to x to get:
Assuming that the head loss can be expressed using Chezy equation, we have:

7. Gradually Varied Flow Equation
Consider the steady non uniform flow in a channel. We wish to develop an equation for the variation of the water surface h(x), i.e. longitudinal water surface profile.
For this, we will consider the equation of energy:
and the equation of continuity:
ref. line
Differentiate the energy equation with respect to x to get:
Assuming that the head loss can be expressed using Chezy equation, we have:

8. Gradually Varied Flow Equation
Note that for a prismatic channel the flow area is only a function of the flow depth, A = f(h) :
We can, therefore, write:
Substitute this expression back into the previous equation to get:
By rearranging the terms, we obtain a differential equation describing the variation of flow depth with distance, i.e. the equation for longitudinal water surface profile:
It is important to note that when:
i.e. Chezy equation for uniform flow
the water surface profile equation reduces to

9. Gradually Varied Flow Equation
The flow depth remains constant and is equal to normal depth (uniform flow)
The flow depth increases in the direction of flow
The flow depth decreases in the direction of flow
Consider again the equation for longitudinal water surface profile:
For
the denominator becomes zero and we have:
We can, therefore conclude that, at critical flow (Fr = 1 and h = hc), the water surface profile is perpendicular to bed.
The normal is equal to critical depth, hn = hc , when:
Uniform flow

10. Review of the Notion of Critical Flow
Consider the flow cases below (for all cases channel cross section characteristics are the same):
Critical slope is the bed slope when normal depth, hn, is equal to critical depth, hc.
When flow is critical, we have:
Since the flow is also uniform, Chezy equation holds:
Equating two expressions, we have:
note that we have changed So to Sc.
The expression for critical discharge is obtained as:
If Manning-Strickler is used:

11. Gradually Varied Flow Equation in Terms of Conveyance
The equation for gradually varied flow can also be written using the notion of conveyance:
when the flow is uniform, in either case we can write:
Remember the definition of conveyance:
when using Manning Strickler or
when using Chezy
Consider the term in the denominator of gradually varied flow equation:
Now consider the term in the nominator of gradually varied flow equation:
The gradually varied flow equation can therefore be written as:

12. Special forms of Gradually Varied Flow Equation: Wide Channel
Let us now consider a wide rectangular channel. The Chezy equation can be written as:
The critical depth in a rectangular channel is given by:
Using these expressions and assuming that the Chezy coefficient C does not depend on depth h, the gradually varied flow equation can be written as:
This equation is known as equation of Bresse
named after the French scientist J.A.C. BRESSE ( ), who developed it first.
If we use Manning-Strickler, we have:
In this case equation of Bresse becomes:

13. Gradually Varied Flow: Forms of Water Surface
Before we present all possible gradually varied flow profiles, let us take a look at the general properties of such curves:
The water surface profile approaches asymptotically to uniform depth hn.
The water surface profile is orthogonal to the critical depth line, when h = hc.
Water surface profiles are classified according to the bed slope.
Channel on Mild slope
type profile
For each profile type several possibilities are distinguished. These are called branches.
Channel on Steep slope
type profile
Channel on Critical slope
type profile
Channel on Horizontal slope
type profile
Channel on Adverse slope
type profile
In studying gradually varied water surface profiles we should also keep in mind that:
In subcritical flow (Fr < 1), the perturbations travel both upstream and downstream. The water surface profiles for subcritical flow are controlled by a downstream control section.
In supercritical flow (Fr > 1), the perturbations travel only downstream. The water surface profiles for supercritical flow are controlled by an upstream control section.

14. Gradually Varied Flow: Forms of Water Surface
Convention for numbering branches:
When the water surface profile is higher than both the normal depth and the critical depth, the branch is numbered as type 1,
the water surface profile is between the normal and critical depths, the branch is numbered as type 2,
the water surface profile is lower than both the normal depth and the critical depth, the branch is numbered as type 3,

15. Gradually Varied Flow: Forms of Water Surface
Channel on Mild slope
and
M-type profiles
Branch M1
Branch M2
Branch M3
Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve tends to become horizontal.
Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve decreasingly tends to critical depth.
Towards downstream the profile approaches increasingly to critical depth.
Encountered:
Upstream of a weir or a dam
Upstream of a pier
Upstream of certain bed slope changes points
Encountered:
Upstream of an increase in bed slope
Upstream of a free drop structure
Encountered:
When a supercritical flow enters a mild channel
After a change in slope from steep to mild

16. Gradually Varied Flow: Forms of Water Surface
Channel on Steep slope
and
S-type profiles
Branch S1
Branch S2
Branch S3
Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve tends to become horizontal.
Towards upstream the profile approaches asymptotically normal depth, towards downstream the curve decreasingly tends to critical depth.
Towards downstream the profile approaches increasingly to critical depth.
Encountered:
Upstream of a weir or a dam
Upstream of a pier
Upstream of certain bed slope changes points
Encountered:
Upstream of an increase in bed slope
Upstream of a free drop structure
Encountered:
When a supercritical flow enters a mild channel
After a change in slope from steep to mild

17. Gradually Varied Flow: Forms of Water Surface
Channel on Critical slope
and
C-type profiles
Branch C1
Branch C2
Branch C3
The water surface profile is horizontal, when Chezy equation is used.
There is no physically possible C2 profile.
The water surface profile is horizontal, when Chezy equation is used.
Encountered:
Upstream of a dam/weir
At certain bed slope change locations
Encountered:
When a supercritical flow enters a mild channel
After a change in slope from steep to mild

18. Gradually Varied Flow: Forms of Water Surface
Channel on Horizontal slope
H-type profiles
Branch H1
Branch H2
Branch H3
Normal depth becomes infinite and is meaningless. Consequently, H1 profile is not possible.
Similar to M2 profile
Similar to M3 profile
Encountered:
Upstream of a free drop structure
Encountered:
When a supercritical flow enters a horizontal channel

19. Gradually Varied Flow: Forms of Water Surface
Channel on Adverse slope
H-type profiles
Branch A1
Branch A2
Branch A3
Normal depth becomes infinite and is meaningless. Consequently, A1 profile is not possible.
Similar to H2 profile (parabolic)
Similar to H3 profile
(parabolic)
Encountered:
Upstream of a certain bed slope change location
Encountered:
When a supercritical flow enters a channel with adverse slope

20. Gradually Varied Flow: Notion of Control Section
Note that the passage from subcritical flow to supercritical flow occurs with a smooth surface.
On the other hand, when the flow passes from supercritical flow to subcritical flow, a sudden increase in the water depth is observed. On the figure this is indicated by HJ, which means hydraulic jump. We will study hydraulic jump in more detail later.
Photograph from Ohio University's Fluid Mechanics Laboratory. Athens, Ohio USA
Control point, as the name implies, is the point that controls the water surface profile. At a control point we can generally write an expression between discharge and depth. Thus, it can be used as boundary condition for calculating the water surface profile.

21. Critical Depth as Control Section and Other Uses of Critical Depth
In open channel flow locally a critical flow situation may exist for certain situations, such as slope change from mild to steep, free fall (drop structure), and excessive contraction, etc.
Changing from a mild slope to a steep slope (passage from subcritical flow to supercritical flow).
Subcritical flow at a free overfall.
In fact, the critical depth takes place about 3 to 4 times hc upstream of the brink (due to curvature of streamlines). The depth at the brink is approximately equal to:
The cases of critical flow due to excessive contraction and a high positive step will be studied later.
In open channel flow critical section is a valuable tool because, knowing the geometry of the section, one can write the relationship between flow depth and discharge.
Due to this property, critical condition is sometimes forced at a point in the channel. Then the discharge can be obtained by measuring the flow depth.

22. Computation of Gradually Varied Flow
Several methods are available for computing gradually varied water surface profiles:
The most obvious is to solve the differential equation of gradually varied flow, equation of Bresse, using a numerical method, such as 4th order Runge-Kutta method. This method is called method of direct integration.
Equation of Bresse using Manning-Strickler equation:
Equation of Bresse using Chezy equation:
4th order Runge-Kutta method formula can be written as:
where:
Coordinate along the channel length. The origin can be arbitrarily placed at any location.
Flow depth at location x. All flow parameters at this location are known.
Flow depth at location x+Dx. This is the unknown flow depth we are calculating.
Computations should start from a point where all flow parameters are known (such as a control point) and proceed upstream if the flow is subcritical and downstream if the flow is supercritical.

23. Computation of Gradually Varied Flow
Several methods are available for computing gradually varied water surface profiles:
The second possibility is to use directly the energy equation to compute the water surface profile by employing an iterative procedure. This approach is called method of successive approximations.
This method can be applied in two ways:
2.1 The open channel reach under study is divided into sub-reaches at known intervals starting from a control point where all the hydraulic parameters are known. Based on the depth at the known point the depth at the next station is computed. This method is called method of reaches (Stand Step Method in Open-Channel Flow, MH Chaudhry).
2.2 A control point where all the hydraulic parameters are known is identified. The depth at that station, h, is known. We choose another depth h+Dh, and compute where this depth will be along the channel. This method is called method of depth variation (Direct-Step Method, MH Chaudhry).
In this course, we will study only the method of reaches.
Please refer to the textbook and other references for more information on other methods that can be used for computation of water surfaces.

24. Computation of Gradually Varied Flow: Method of Reaches
Consider the gradually varied flow shown in the figure.
We have divided the reach under study into smaller sub reaches of length Dx. We also define the cross sections i, i+1, i+2, ….. etc.
We will assume that the geometric properties of the channel (A, P, B, Rh, Dh) at each cross section can be calculated by knowing the depth.
We will also assume that the depth at cross section i is known. We would like to calculate the depth at cross section i+1.
Let us write the equation of energy Bernoulli equation) between two cross sections i and i+1 :

25. Computation of Gradually Varied Flow
Therefore, when using the method of reaches, we will be solving this ordinary differential equation:
The basic equation we are using is:
Since depth hi , invert elevation zi and the discharge Q are known, we can calculate the left side of the equation, i.e. the total energy head, Hi directly.
Let us now assume a depth hi+1 . Since the invert elevation zi and the discharge Q are known, we can also calculate the total energy head, Hi+1 directly.
Now the question is weather the assumed that is the correct depth. This can be easily done. If the assumed depth hi+1 is correct, then, the difference between the total heads Hi and Hi+1 should be equal to Dx Se.
The energy gradient can be calculated using either the equation of Chezy or Manning Strickler:
Chezy equation:
Manning Strickler equation:

26. Computation of Gradually Varied Flow
Since the hydraulic parameters are varying from cross section i to i+1, we may want to use the average value of the energy gradient:
Note also that :
We should therefore check that: or
is satisfied.
If the above equation is not satisfied, a new value should be assumed for hi+1 and the computations must be carried out again.
All these calculation can easily be carried out on a spread sheet.
If there are singular losses between the two cross sections i and i +1, this should also be taken into account. Then the equation becomes:
Again considering average values we can write:
Consequently:

27. Computation of Gradually Varied Flow
A trapezoidal channel having a bottom width of b = 7.0m and side slopes of m = 1.5, conveys a discharge of Q = 28m3/s. The channel has a constant bed slope of So = The Manning friction coefficient for the channel is n = 0.025m-1/3s. The channel terminates by a sudden drop of the bed.
Determine the type of water surface profile to be expected.
Calculate the water surface profile for a reach length of 3200m.
The computation of gradually varied flow equations can be easily carried out on a spreadsheet using Goal Seek function

28. Rapidly Varied Flow at Channel Transitions
A transition is a change in the channel geometry over a relatively short distance. The change can be contraction or expansion of the section, or a change in the section cross section geometry (say from rectangular to trapezoidal), or an abrupt rise or drop of the channel bed. In designing transition, the attention must be paid to create minimum amount of disturbance to the flow.
The figure shows, typical designs for channel transition from a rectangular cross section to a trapezoidal cross section.

29. Use of Specific Energy to study Rapidly Varied Flow at Channel Transitions
Curve plotted for a constant Q
Subcritical flow
Specific Energy
Alternate depths
Supercritical flow
Specific energy curve is an extremely useful tool for analyzing various flow situations. In the following slides we will learn how the specific energy curve can be used to analyze various flow situations in channel transitions (flow over a positive or negative step, flow through a contraction or expansion).

30. Rapidly Varied Flow at Channel Transitions
In the following pages we will study in detail the rapid change of water surface at four types of channel transitions under both subcritical and supercritical conditions:
Subcritical flow over a positive step
Supercritical flow over a positive step
Subcritical flow over a negative step
Supercritical flow over a negative step
Subcritical flow through a contraction
Supercritical flow through a contraction
Subcritical flow through an expansion
Supercritical flow through an expansion
Side view
Side view
Top view
Top view

31. Rapidly Varied Flow: Subcritical Flow over a Positive Step
Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).

32. Rapidly Varied Flow: Supercritical Flow over a Positive Step
Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).

33. Rapidly Varied Flow: Subcritical Flow over a Negative Step
Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).

34. Rapidly Varied Flow: Supercritical Flow over a Negative Step
Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).

35. Rapidly Varied Flow: Subcritical Flow through a Contraction
Top view
Side view
Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).

36. Rapidly Varied Flow: Supercritical Flow through a Contraction
Top view
Side view
Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).

37. Rapidly Varied Flow: Subcritical Flow through an expansion
Top view
Side view
Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).

38. Rapidly Varied Flow: Supercritical Flow through an Expansion
Top view
Side view
Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).

39. Rapidly Varied Flow: Special Case of Choked Flow due to a High Positive Step
By subtracting Dz from Hs1, we cannot fall back onto the specific energy curve
At one point on the step the flow goes through the critical depth of the cross section on the step.
final energy line
initial energy line
final water surface
initial water surface
If the step is too high, subtracting Dz from Hs1, we cannot fall back onto the specific energy curve. The flow is said to be choked. The step is too high. The water accumulates upstream of the step until it can pass over it by going through critical flow over the step. Same equations hold. However, now h1 is also an unknown. Condition of critical flow over the step provides the third equation needed for the analysis.

40. Rapidly Varied Flow: Special Case of Choked Flow due to too much Contraction
Top view
With the available energy Hs1, we cannot cut the specific energy curve of the contracted section
At one point on the contracted section the flow goes through the critical depth of that cross section.
final energy line
initial energy line
final water surface
initial water surface
Side view
If the step is contracted too much, with the specific energy Hs1 we cannot cut the specific energy curve of the contracted section. The flow is said to be choked. The section is contracted too much. The water accumulates upstream of the contraction until it can pass a discharge of Q to the downstream by going through critical flow of the contracted section. Same equations hold. However, now h1 is also an unknown. Condition of critical flow at the contracted section provides the third equation needed for the analysis.

41. Energy Losses For Subritical Flow in Open Channel Transitions
Head losses at contractions and expansions can be calculated using the following expressions: U1 U2 U1 U2
Contraction
Expansion
Taken from USACE (1994)
U.S. Army Corps of Engineers, “Hydraulic Design of Flood Control Channels,” Engineering and Design Manual, EM , July 1991, Change 1 (June 1994).

42. Assuming a rectangular channel, we have:
Rapidly Varied Flow: Hydraulic Jump
Hydraulic jump is a natural phenomenon that occurs when supercritical flow is forced to become subcritical.
The passage from supercritical flow to subcritical takes place with a sudden rise of the flow depth accompanied by a very turbulent motion that may entrain air into the flow.
To derive the equation governing hydraulic jump in a channel (see figure above), we will make use of momentum and continuity equations simultaneously.
Consider a control volume, which comprises the hydraulic jump. The upstream cross section of the control volume is in supercritical flow and the downstream section is in subcritical flow. Forces acting on this control volume are the weight of the fluid, W, the upstream and downstream pressure forces, FP1 and FP2 respectively, and the friction force, Ff. The momentum equation can be written as:
Assuming a rectangular channel, we have:

43. Divide both sides by (h2 – h1) to get:
Rapidly Varied Flow: Hydraulic Jump
Using these expressions and neglecting the component of weight and friction forces, the momentum equation becomes:
Note that the left and right hand side of the equation represent the specific momentum, which is defined as:
Let us now make use of equation of continuity to write the momentum equation as:
Divide both sides by (h2 – h1) to get:
Only the positive root of the above quadratic equation is physically meaningful:
Written in dimensionless form, the above equation becomes: or
This equation is called the equation of Bélanger in honor of the French scientist who developed it for the first time.
where
and

44. Rapidly Varied Flow: Hydraulic Jump
Note that for a hydraulic jump on larger slopes, the weight of the fluid cannot be neglected. In this case, the equation of Bélanger for hydraulic jump becomes:
where
as given by Rajaratnam.
a is in degrees
Hydraulic jumps are classified according to the approach flow Froude number.

45. Energy loss across the hydraulic jump:
Rapidly Varied Flow: Hydraulic Jump
Photos from (Dr. H. Chanson):
Energy loss across the hydraulic jump:
Length of the hydraulic jump:
Classification of hydraulic jumps:
Undular jump
Weak jump
Oscillating jump
jump type generally preferred in engineering applications
Steady jump
Strong jump

46. Use of Hydraulic Jump in Hydraulic Engineering
Hydraulic jump is used for dissipating the energy of high speed flow which may harm the environment if released in an uncontrolled way.
The hydraulic jump, should take place in a area where the bottom is protected (for example by a concrete slab or large size rocks). If the jump takes place on erodible material the formation of the erosion hole may endanger even the foundations of the structure.
In real engineering projects measures are taken to ensure that the hydraulic jump takes place in the area with a protected bottom. This is achieved by creating a stilling basin with the use of chute blocks, baffle piers, and end sill, etc.

47. Examples of the Use of Hydraulic Jump in Hydraulic Engineering

48. USBR Type I Stilling Basin USBR Type II Stilling Basin
Design of Stilling Basins
USBR Type I Stilling Basin
USBR Type II Stilling Basin
USBR Type III Stilling Basin
USBR Type IV Stilling Basin
SAF Stilling Basin
Pillari’s Stilling Basin

49. Hydraulic Design of Stilling Basins and Energy Dissipators
Books on Design of Stilling Basins
Hydraulic Design of Stilling Basins and Energy Dissipators
by A. J. Peterka, U.S. Department of the Interior, Bureau of Reclamation
Energy Dissipators and Hydraulic Jump
by Willi H. Hager

50. Thus, the jump takes place between A and Z.
Positioning of a Hydraulic Jump
Draw the downstream subcritical flow profile starting from a control section at the downstream.
Draw the upstream supercritical flow profile starting from a control section at the upstream.
Draw the conjugate depth curve for the upstream supercritical flow profile.
For a hydraulic jump with zero length the jump is a vertical water surface between A’ and Z’.
If we wish to take into account the length of the jump for each point on the conjugate depth curve, draw a line parallel to the bed. The length of the line should be equal to the length of the jump, i.e. 3 to 5 times the height difference between the conjugate depth and the water depth. The tips of these lines are joined to obtain a translated conjugate depth curve which takes into account the length of the jump. The intersection of the downstream profile with the translated conjugate depth gives the downstream end of the jump.
Thus, the jump takes place between A and Z.

51. Specific Energy Consider again the specific energy curve.
Oblique Hydraulic Jump
Consider again the specific energy curve.
Curve plotted for a constant Q
Subcritical flow
Specific Energy
Alternate depths
Supercritical flow
It can be seen that when the flow is supercritical, a small variation in depth (say Dh) causes a large variation in kinetic energy and, thus the specific energy (DEs or DHs).
Therefore, in supercritical flow, a transition, such as a change in width or a change in direction, will provoke an abrupt variation of flow depth and stationary, stable gravity waves will appear on the free surface.

52. Oblique Hydraulic Jump
Referring to the figure on the left, consider the case of a supercritical deflected by a side wall making an angle q with the approach channel.
A standing wave front forms making an angle b with the approach channel direction. This is called an oblique (hydraulic) jump. Note that this is somewhat different than a classical hydraulic jump due to the fact that the flow is still subcritical downstream of the wave front:
and
The continuity equation in the direction normal to the wave front gives:
Neglecting the bottom friction, the momentum equation in the direction normal to the wave front gives:
Froude numbers using the velocity components normal to the oblique wave front are defined as:
and

53. This clearly shows that:
Oblique Hydraulic Jump
Note that in the direction tangent to the wave front no momentum change takes place. The equation of momentum in tangential direction becomes:
This clearly shows that:
Geometric considerations allow us to write:
Combining continuity and momentum equations, one obtains the equation for change of depth across an oblique jump:
In which the Froude number normal to the wave front is defined as:

54. The angle b of the wave front can be expressed as:
Oblique Hydraulic Jump
The angle b of the wave front can be expressed as:
Note that, for small variations of depth, thus for gradual transitions, one gets:
Using equation of continuity and geometric relationships, the equation for the oblique jump can also be written as:
Combining two equations for change of depth across an oblique jump, we can write:
These derivations were originally carried out by Ippen (1949). He also experimentally verified the relationship above which gives the relationship between q and b for contracting channels (only).

55. Asymmetrically converging channel, and
Oblique Hydraulic Jump
The relation ship between q and b is plotted on the left. Following observations can be made:
For all Froude numbers there exists a maximum value for the angle of deflection, qmax.
For all values q smaller than qmax, two values of b are possible. However, since the analysis is made for the case the flow remains supercritical after the jump, i.e. Fr2 > 1, we should consider only the values on the left side (solid lines).
It is important to note that, any perturbation created by one wall will be reflected by the other wall and so on. To study this behavior, we will consider two cases:
Asymmetrically converging channel, and
Symmetrically converging channel.

56. Reflection of Oblique Jumps in an Asymmetrical Channel Contraction
Consider the channel on the left. The left wall is deflected into the flow by an angle q, while the right wall remains straight.
… and so on. If the contracting channel is sufficiently long, the wave reflection continues until the flow finally becomes subcritical.

57. Reflection of Oblique Jumps in an Asymmetrical Channel Contraction
Consider the channel on the left. The left wall is deflected into the flow by an angle q, while the right wall remains straight.
… and so on. If the contracting channel is sufficiently long, the wave reflection continues until the flow finally becomes subcritical.

58. View of Supercritical Flow in Curved Transition:
Reflection of Oblique Jumps in an Symmetrical Channel Contraction
View of Supercritical Flow in Curved Transition:
Experimental Visualization of Standing Wave Patterns, after Ippen and Dawson (1951);
Computer Visualization Based on 2D, Shock-Capturing, Numerical-Model Predictions
Causon D. M., C. G. Mingham and D. M. Ingram (1999), Advances in Calculation Methods for Supercritical Flow in Spillway Channels, ASCE, Journal of Hydraulic Engineering, Vol. 125, No. 10, pp

59. From geometric considerations , we also have the relationship:
Designing a Symmetrical Channel Contraction for Supercritical Flow
A good channel contraction design for supercritical flow should reduce or eliminate the undesirable cross wave pattern. This can be achieved by choosing a linear contraction length LT, thus by choosing a contraction angle q’, such that the positive waves emanating from points A and A’, due to converging walls, arrive directly at points D and D’, where negative waves are generated due to diverging walls. Such a design is shown in the figure on the left.
The choice of the angle q’ depends on the approach Froude number, Fr1, and the contraction ratio B3/B1.
Based on continuity equation, and assuming that the flow remains supercritical in the contracted section, i.e. Fr3 > 1, we can write:
From geometric considerations , we also have the relationship:
The angle q’ to satisfy these two equations is calculated using an iterative procedure.

60. The continuity equation with lateral flow becomes:
Gradually Varied Flow with Lateral Inflow
We will now consider the case of gradually varied flow with lateral inflow. In the most general case, the discharge added subtracted laterally affects both to the mass and the momentum of the flow.
Consider the gradually varied flow with lateral inflow as shown on the left.
The continuity equation with lateral flow becomes:
represents lateral discharge which can be positive if a discharge is added, or negative if a discharge subtracted.
We can also write:
Equation of the momentum states that the sum of all forces is equal to the change in momentum:
Let us now analyze all the terms in this equation one by one.

61. The net hydrostatic pressure force will be:
Gradually Varied Flow with Lateral Inflow
The net hydrostatic pressure force will be:
zP is the distance from the free surface to the centroid of the flow area A:
The weight of the water prism between two sections that are dx apart is:
The rightmost side assumes that a is small.
The friction force can be written as:
since
The change in the momentum can be written as:
Note that
Note also that the lateral flow is entering or leaving the channel at an angle f and with velocity Uℓ.
Let us insert above expression the equation of momentum, and simplify:

62. Gradually Varied Flow with Lateral Inflow
This is the equation of free surface for a steady gradually varied flow with lateral inflow, which is also called a steady spatially varied flow.
Recalling that:
and
Simplifying also second order terms AdA and dQdA, the spatially varied flow equation can be written as:
It can be verified that, this equation reduces to gradually varied flow formula when lateral flow is zero, qℓ = 0.
Spatially varied flow equation can also be solved using the same methods for solving gradually varied flow equation.

63. Side channel spillway of Hoover Dam in Nevada.
Example of Structures for Spatially Varied Flow: Side Channel Spillway
Side channel spillway of Hoover Dam in Nevada as seen from the reservoir side.
Side channel spillway of Hoover Dam in Nevada.

64. Side channel spillway of Hoover Dam in Nevada, looking downstream.
Example of Structures for Spatially Varied Flow: Side Channel Spillway
Side channel spillway of Hoover Dam in Nevada, looking downstream.

65. Quiz No 1(5 minutes)
A steep channel is connected to a mild channel as shown in figure. Both channels have a rectangular cross section. The following data is given:
The flow in steep channel is steady and uniform. Determine in which channel, steep or mild, the hydraulic jump will take place.

66. Quiz No 1(5 minutes): Solution
A steep channel is connected to a mild channel as shown in figure. Both channels have a rectangular cross section. The following data is given:
The flow in steep channel is steady and uniform. Determine in which channel, steep or mild, the hydraulic jump will take place.
Solution: Assume that the steady uniform flow continues all the way down to the point where the slope becomes mild. Let us see if there is a jump at that point what would be the conjugate depth.
A jump taking place at the point of slope change will be too strong. It can jump higher than the normal depth in the channel. Therefore, the flow continues into the mild channel without a jump and creates an M3 type profile up to a depth whose conjugate depth is equal to the uniform flow depth of the mild channel.

67.   Consider the channel on the left with the following data:
Quiz No 2 (5 minutes)
Consider the channel on the left with the following data:  
Determine if the flow is choked due to the positive step. What will be the flow depth over the step?

68.   Consider the channel on the left with the following data:
Quiz No 2 (5 minutes): Solution
Consider the channel on the left with the following data:  
Determine if the flow is choked due to the positive step. What will be the flow depth over the step?
Solution: Assuming no singular energy losses due to the step, the energy grade line remains at the same level. Over the step, the energy is reduced by an amount Dz.
Thus the flow is choked. The flow will go through the critical depth over the step, i.e.