Hydraulic Engineering: Calculating Boundary Shear Stress in a Trapezoidal Channel

How can we calculate the boundary shear stress in a trapezoidal channel based on the given data?

Calculate the average boundary shear stress and the maximum boundary shear stress in a trapezoidal channel with specific characteristics.

Calculation of Boundary Shear Stress in a Trapezoidal Channel:

To calculate the boundary shear stress in a trapezoidal channel, we first need to determine the average boundary shear stress and the maximum boundary shear stress. The average boundary shear stress can be calculated using the formula τb = (γ × R × S) / n, where γ represents the unit weight of water, R is the hydraulic radius, S is the longitudinal bed slope, and n is the Manning's coefficient.

a) Average Boundary Shear Stress:

The formula for the average boundary shear stress is τb = (γ × R × S) / n. By calculating the hydraulic radius, we can determine the average boundary shear stress value.

b) Maximum Boundary Shear Stress:

The maximum boundary shear stress can be found using the formula τmax = τb × Kc, where Kc is the coefficient of contraction.

c) Maximum Discharge Capacity:

If the mean diameter of the material forming the channel bed and sides, along with the angle of repose, are given, the maximum discharge that can pass through the channel without causing scour can be determined.

Calculation Details:

For a trapezoidal channel with specific characteristics such as bed width, side slope, longitudinal bed slope, mean velocity, and Manning's coefficient, we can calculate the boundary shear stress as follows:

The hydraulic radius (R) can be determined using the cross-sectional area (A) and the wetted perimeter (P) formulas. The values of R, A, and P are calculated based on the given data, leading to the calculation of the average boundary shear stress.

The coefficient of contraction (Kc) is used to calculate the maximum boundary shear stress. Understanding the maximum boundary shear stress provides insights into the channel's flow behavior and stability.

Considering the material characteristics and angle of repose, the maximum discharge that the channel can handle without causing scour can be determined. This information is crucial for designing and managing water flow in the channel effectively.