Hydrodynamic Concepts River Mechanics CH-2 Hydrodynamic Concepts

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 =

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

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

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?

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

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

Energy Equation Difference in total head between two sections:

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

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

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

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

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

Specific Energy

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

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

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