Hydrodynamic Concepts

Presentation on theme: "Hydrodynamic Concepts"— Presentation transcript:

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