Lecture 3 - Depth Vs Discharge Vs Resistance In Open Channels (Uniform Flow)

What are the Key Relationships for Open Channel Flow in Uniform Conditions?

Depth and Velocity are constant with position (therefore no acceleration)

• Friction Slope (S_f) = Bed Slope (S_o) = Water Surface Gradient

What are the Flow Resistance Equations and What Assumptions are made?

Uniform Flow, hence no acceleration and forces must be balanced 

Weight = Frictional Resistance from boundary

• T₀ = Bed Shear Stress

• P = Wetted Perimeter

• L = Length

• p = density 

• g = gravity

• A = Area

• x = Slope Angle

• Assuming shallow slopes and that shear stress is proportional to V²

• V = Velocity   

• C_h = Chezy C    

• S₀ = Bed/Energy Slope

Why is Chezy C viewed as unreliable?

As Chezy C is found to be highly variable with velocity and boundary roughness (Friction)

What is Manning's n based on and What is the Gauckler-Manning Formula used for?

Mannings n is based only on the roughness of the boundary

Gauckler Manning Formula is used to calculate Velocity and Discharge

• R = Hydraulic Radius    

• n = Manning’s n    

• P = Wetted Perimeter

What is the assumption used in the Gauckler-Mannings Formula?

Assuming boundary friction is the only resistance force, in these cases Mannings n is assumed to be constant with depth

What are other potential things that generate Resistance and What Assumption do we use for these cases?

• Sinuosity

• Channel obstructions

• Vegetation

• Irregular cross sections

In these cases we assume manning’s n is not constant with depth

Why is Calibration necessary when using Mannings n in Open Channels?

Empirical coefficients are prone to inaccuracies

How is Roughness adjusted depending on the Situation?

• Frictional coefficients can be adjusted to take into account resistance from other sources

• In many cases effects are non uniform with depth

• Roughness varies with season

What are the issues when the Channel has a Distorted Cross-Section?

• Small rise in water can lead to large changes to wetted perimeter

• Boundary friction effects are less uniform

• Calibrating Manning’s n must be calculated for each section

How do we handle Distorted Cross-Section Channels?

• As Velocity and Manning’s n may be different for the main channel and sides

• Cross section is divided into sections and the flow is considered to be composed of flow in parallel channels

• Friction effects assumed negligible along boundaries between sections

How do we calculate the Area and Wetted Perimeter for a Rectangular Shaped Channel?

Area

Wetted Perimeter

• B = Width

• y = Depth
  

How do we calculate the Area and Wetted Perimeter for a Triangular Shaped Channel

Area

Wetted Perimeter

Z = Change in horizontal when going down vertically by 1

How do we calculate the Area and Wetted Perimeter for a Trapezoidal Shaped Channel?

Area

Wetted Perimeter

How do calculate the Velocity and Discharge of a Full Circular Conduit?

• r = Radius

How do we calculate the Velocity and Discharge for a Partially Full Circular Conduit?

• Calculate the velocity and discharge assuming the conduit is full

• Then multiply velocity and discharge by a factor received from the Circular Section - Proportional Velocity and Flow Curves