1. Hydrodynamic Concepts
River Mechanics CH-2
Hydrodynamic Concepts

2. Continuity Equation
Consider the cross-section below of a prismatic channel:
Equation of continuity relates to changes in volume of water within a cross-section (top-width, B, and area, A) of length dx:
Entering Volume =
Volume Leaving =

3. Continuity Equation What is the change in volume (DV)?
Change in volume in cross-section of length dx can also be expressed as…

4. Continuity Equation We can equate these two DV equations:
If you have lateral inflow/outflow (ql):

5. Continuity Equation We can write this equation in many forms:
Steady with no inflow/outflow?
Non-steady expressed in terms of U?
Non-steady expressed in terms of U and Dh?

6. Energy Equation Usually write Bernoulli’s equation: Assumptions:
Steady Flow
Pressure is hydrostatic – when is the pressure not hydrostatic?
Weak channel
Fluid particles move at the same velocity, U – kinetic energy correction factor (ae = 1 for turbulent flow)
Flow is one-dimensional
Incompressible fluid

7. Energy Equation For a real fluid… Energy or head loss due to friction:
P = perimeter of the end area, A

8. Energy Equation
Difference in total head between two sections:

9. Energy Equation
So what is the equation for the difference in energy?

10. Energy Equation
We can simplify the equation just written to derive a fundamental equation for unsteady, non-uniform flow:

11. Energy Equation
How does this equation simplify for steady, non-uniform flow?
How does this equation simplify for steady, uniform flow?

12. Saint-Venant Equations
Combined equation of continuity and equation of energy…

13. Specific Energy Plane of reference at bed slope (Sf):
Specific Energy, Hs –velocity and pressure heads, fraction of the total head

14. Specific Energy

15. Critical Depth
Solved for by minimizing the specific energy…
Fr=1:

16. Specific Energy For alternate depths…
h < hc – Supercritical (Torrential) Regime
h > hc – Subcritical (Fluvial Regime)
A steady flow in a channel can exist in two ways:
Supercritical regime – depth is small and velocity large
Subcritical regime – depth is large and velocity small

17. Discharge Curve Using equation for specific energy…
Curve is maximized when dQ/dh = 0…
Value of h that corresponds to the maximum discharge is the critical depth, hc