Open Channel Flow - Civil Engineering SSC JE (Technical) - Civil Engineering

= velocity at a depth of 0.8 y₀ from the free surface.

Terms related to open channel flow

• Depth of flow (y) : Vertical distance between the lowest point of the channel section (the bed) and the free liquid surface.
• Top width (T) : Width of the channel section at the free liquid surface.
• Wetted area (A) : Cross-sectional area of flow, measured normal to the direction of flow.
• Channel slope (S) : Inclination of the channel bed, given by S = tan α ≈ sin α = h/l, where h is the vertical fall over length l of channel.
• Wetted perimeter (P) : Length of the channel boundary (sides and bed) in contact with water.
• Hydraulic radius (R) : R = A / P.
• Hydraulic depth (D) : D = A / T.
• Hydraulic grade line (HGL) : Line representing pressure head along the channel. For open channel flow it coincides with the free surface.
• Energy grade line (EGL) or total energy line (TEL) : Line representing total energy (per unit weight) of the liquid with respect to a datum. The slope of the TEL is the hydraulic slope. The difference between TEL/EGL and HGL at any section equals the velocity head multiplied by the kinetic energy correction factor.

• Difference between TEL and HGL at any section =
  
  where a is the kinetic energy correction factor.

Specific energy

• The total energy of a section of channel flow referred to a datum is the sum of elevation head, pressure head and velocity head.
• If the datum is taken at the channel bed at the section, the resulting expression is called the specific energy E (specific energy measured with respect to the channel bed):

If datum is the channel bed, specific energy E = y + α V²/(2g).

• For a uniform flow the specific energy remains constant along the channel. For varied flow the specific energy may increase or decrease.
• If frictional resistance is neglected in non-uniform flow, the total energy will be constant but the specific energy will be constant only for a horizontal channel; for other cases specific energy varies.

Critical depth

• For a given discharge Q in a channel, the specific energy E as a function of depth y has a characteristic shape. For some Q there are two positive depths that give the same specific energy; these are called alternate depths.
• At the minimum of the E-y curve the two alternate depths merge. The minimum specific energy is called the critical specific energy E_c and the corresponding depth is the critical depth y_c. For E < E_c flow is not possible.
• At the critical depth the specific energy is minimum and flow is said to be in a critical state.

Equations (i) & (ii) are the basic expressions for critical flow conditions in a channel (see figure above).

Froude number

• The Froude number F is defined as:

where D = A / T is the hydraulic depth. At critical flow F = 1 when y = y_c.

• For alternate depths in the same channel:
• y₁ < y_c : F₁ > 1 ⇒ supercritical flow
• y₂ > y_c : F₂ < 1 ⇒ subcritical flow

Uniform flow

• Uniform flow occurs when depth, area and velocity remain constant along the length of a prismatic channel (a channel with constant cross-section and slope).
• In uniform flow the total energy line, channel bottom and water surface are all parallel.

Velocity formulae for uniform flow

Chezy equation

• Chezy formula: V = C √(R S) where

V = mean velocity of flow

R = hydraulic radius = A / P

S = bed slope (for uniform flow S ≈ energy slope)

C = Chezy coefficient (dimension [L1/2 T-1]); C depends on surface roughness and flow conditions.

• C is sometimes made dimensionless by dividing by √g. C is also called Chezy's coefficient.

Relationship between Chezy coefficient C and friction factor f

• For pipe flow the Darcy-Weisbach equation applies for head loss due to friction.

where h_f = head loss due to friction, D = pipe diameter, f = Darcy friction factor, L = length of pipe.

• For circular conduits the hydraulic radius R = A / P is related to diameter; for a full pipe R = D/4.

Manning's formula

where n is the Manning roughness coefficient. The dimension of n is [L-1/3 T]. Manning's formula is widely used for uniform flow calculations in open channels.

• Comparing Chezy and Manning forms gives relations between n, C and f. The comparison produces the standard transformation shown below.

Ganguillet and Kutter formula

This is another empirical velocity formula used in practical hydraulic design; it uses a roughness coefficient similar in role to Manning's n.

Hydraulically efficient channel sections

The hydraulically efficient section is the channel cross-section which conveys the maximum discharge for a given wetted perimeter (or minimum perimeter for a given area), slope and roughness. It is sometimes called the best section.

(a) Rectangular section

For a rectangular section A = B y (B = bottom width). For a given area the section giving maximum discharge occurs for particular proportions; for an efficient rectangular channel the ratio typically yields y = B/2 and R = y/2 (derived from optimisation of R = A/P).

(b) Trapezoidal section

A = (B + n y) y = constant

• Hydraulic radius R = y / 2 (for the efficient trapezoidal case shown).
• The efficient trapezoidal channel should correspond to a part of a regular hexagon.
• Side slopes should be 60° with the horizontal in the ideal (best) trapezoidal section.
• Half the top width equals one of the sloping sides in the best trapezoidal geometry.

(c) Triangular section

• For an equilateral triangular (efficient) channel the side slope q = 45°.
• Hydraulic radius R = y / (2√2) for the efficient triangular section

(d) Circular section

Two practical efficiency conditions are commonly given:

• Case 1 - Condition for maximum discharge: y ≈ 0.95 D and R ≈ 0.29 D.
• Case 2 - Condition for maximum mean velocity: y ≈ 0.81 D and R ≈ 0.30 D.

Gradually varied flow (GVF)

• Basic assumptions for GVF analysis:
• The pressure distribution at any section is hydrostatic.
• The resistance to flow at any section may be represented by the corresponding uniform flow formula (for example Manning's), using the energy slope (S_e) rather than the bed slope.

Dynamic equation of GVF

The dynamic equation (differential equation of gradually varied flow) relates the rate of change of depth dy/dx to channel geometry, energy slope, critical depth and other flow parameters. Using the governing equation one classifies water surface profiles according to bed slope and relative positions of depth, critical depth and normal depth.

Classification of water surface profiles

Water surface profiles are named by a letter (indicating bed slope category) and a number (indicating whether depth is above, at, or below critical depth), where the bed slope categories are:

• M Mild slope : y_n > y_c
• C Critical slope : y_n = y_c
• S Steep slope : y_n < y_c
• H Horizontal : S₀ = 0
• O Adverse (backward) slope : S₀ < 0

Where y_n is the normal depth for the given discharge and channel slope, and y_c is the critical depth.

Rapidly varied flow (RVF)

Rapidly varied flow occurs where the depth changes abruptly over a short distance; frictional effects are small in the region of change. A steady example of RVF is the hydraulic jump.

Hydraulic jump

A hydraulic jump is formed when a supercritical flow meets a subcritical flow and the depth abruptly rises from the supercritical depth to the subcritical sequent depth. The process generates large eddies and a surface roller and dissipates a significant portion of the flow energy.

• The upstream point where the jump begins is called the toe of the jump.
• The distance between the start and end of the jump is called the length of the jump L_j.
• The depths at the upstream and downstream ends of the jump are called the sequent depths y₁ (upstream) and y₂ (downstream).
• The specific force diagram for a channel can be used to determine the sequent depths for a given discharge in a horizontal channel.

Condition for critical flow

Hydraulic jump in a rectangular channel

• Sequent depth ratio (relation between y₁ and y₂) is obtained from continuity and momentum (specific force) equations:

(a) The algebraic relations and derivations are shown in the figure below.

• The upstream Froude number F₁ is used to characterise the jump.

Energy loss and jump characteristics

• Energy loss across the jump: EL = E₁ - E₂.

• Height of jump = y₂ - y₁.
• Length of jump L_j ≈ 5 to 7 times the height of the jump (empirical range).
• Ratio of energy loss to initial energy is given by the appropriate formula shown below.

Classification of hydraulic jumps

Type	F1	Remarks / Image
Undular jump	1 - 1.7	EL / E1 ≈ 0
Weak jump	1.7 - 2.5
Oscillating jump	2.5 - 4.5
Steady jump	4.5 - 9
Strong turbulent (choppy) jump	> 9.0

Surges

• A surge is an example of unsteady rapidly varied flow. It travels downstream as a transient wave of change in flow depth and/or discharge.
• Surges occur when there is a sudden change in discharge or depth, for example when a gate is suddenly closed or opened.
• A surge that causes an increase in depth is called a positive surge.
• A surge that causes a decrease in depth is called a negative surge.
• Positive surges are relatively stable and have steep fronts; negative surges are unstable and their form changes as they advance.