Open channel hydraulics CE 3372 – Lecture 13

Outline Flow Terminology Energy Equation Critical Depth/Flow Flow Profiles for GVF Manning’s Equation Understand relationships

Open Channel Design Concepts Interest to engineers: Water surface elevation (WSE) (minimize impact/reduce floods) Discharge –Depth relationships Channel design

Conduits whose upper boundary of flow is the liquid surface. Open Channels Conduits whose upper boundary of flow is the liquid surface. Canals, streams, bayous, rivers are common examples of open channels. Storm sewers and sanitary sewers are special cases of open channels; in some parts of a sewer system these channels may be operated as pressurized pipes, either intentionally or accidentally. Similar to closed, there’s multiple dimensions, but 1D is simplest to analyse.

TYPES OF FLOW Steady Flow – flow, depth and velocity may differ from point to point but remain constant over time Unsteady Flow – flow, depth, and velocity is a function of time Uniform Flow – occurs in prismatic channels when flow depths are equal no change in velocity within the channel: Q, y, A, S are all constant Non-uniform Flow – velocity is not the same at every point Temporal the friction slope Sf (slope of the EGL) would be the same as the bottom slope S0. Draw on board the diagram Steady+unsteady are temporal with time!! And uniform is spatial!! Spatial

Open Channel Nomenclature Flow depth is the depth of flow at a cross-section measured from the channel bottom. y

Open Channel Nomenclature Elevation of the channel bottom is the elevation at a cross- section measured from a reference datum (typically MSL). y z Datum

Open Channel Nomenclature Slope of the channel bottom, So, is called the topographic slope or channel slope. y z So 1 Datum

Open Channel Nomenclature Slope of the water surface is the slope of the HGL, or slope of WSE (water surface elevation). HGL Swse y 1 z So 1 Datum

Open Channel Nomenclature Slope of the energy grade line (EGL) is called the energy or friction slope. EGL HGL Sf V2/2g 1 Q=VA Swse y 1 z So 1 Datum

Steady NON-uniform Flow Based on cross-sections: Section 1 is upstream Section 2 is downstream Sketch of steady flow in a channel

Steady Flow The energy grade line (EGL) is: z_head + P_head + v_head The hydraulic grade line (HGL) is at the water surface Energy grade line (EGL) Hydraulic grade line (HGL) velocity Profile grade line is the channel bottom Sketch of steady flow in a channel pressure (depth) The head loss is depicted as the difference between a horizontal zero-loss energy grade line and the energy grade line elevation PRESSURE IS CALLED DEPTH!!!!!!!

Like closed conduits, the various terms are part of mass, momentum, and energy balances. Unlike closed conduits, geometry is flow dependent, and the pressure term is replaced with flow depth.

Energy relationships Energy equation for closed conduits Energy equation for open conduits There is usually a kinetic energy correction factor (alpha) before the velocity terms but we assume that = 1 for turbulent flow

Specific energy The sum of the depth of flow + velocity head (Head relative to the channel bottom) For a given discharge, the SE can be calculated for various flow depths including critical depth SE is important to find flow depths

Occurs when dE/dy = 0 and Fr = 1 Critical depth Depth of flow for a given discharge, where the specific energy is at a minimum Occurs when dE/dy = 0 and Fr = 1 It is important to calculate yc in order to determine if the flow in the channel will be subcritical or supercritical Can be found through Specific Energy Diagram What? Why is it important? How to find it?de/dy = 0 = KE = PE

Plug & chug. Solve for y 3 roots –1 negative = 2 depths There are 3 flows here. Increase to the right top part of the curve approaches the E = y line and the bottom part of the curve tends toward the x-axis. critical energy or minimum energy, Ec and the corresponding critical depth value, yc The critical depth has a Froude number equal to one and corresponds to the minimum energy a flow can possess for a given discharge. IMP NOTE: Alternate depths for a single specific energy critical depth value mentioned in the E–y diagram section above is mathematically represented by the ratio of the fluid velocity to the velocity of a small amplitude gravity wave. Alternate Depths: Plug & chug. Solve for y 3 roots –1 negative = 2 depths A = By y Q=qy where q is the discharge/unit width of channel B

Open channel flow is also classified by the Froude number Open channel flows Open channel flow is also classified by the Froude number Critical depth, yc occurs at Fr = 1 critical depth value mentioned in the E–y diagram section above is mathematically represented by the ratio of the fluid velocity to the velocity of a small amplitude gravity wave.

Open channel flows Subcritical flow Low velocities, Fr < 1 Disturbance travels upstream y > yc Supercritical flow High velocities, Fr > 1 Disturbances travel downstream y < yc Arbitrary cross sections are handled by numerical integration, regular geometries are considerably simpler.

Arbitrary cross-section Critical Flow T dy Has a minimum at yc y A dA P dE/dy for a minimum, gradient must vanish. Yellow = variation of energy with respect to depth in discharge form More general definition of Fr

Critical Flow – Rectangular channel yc Ac Only for rectangular channels! Given the depth we can find the flow!

Critical Flow: Rectangular Channels velocity head = 0.5 (depth)

Open channel flows Similar to pipe flow, open channel flow can be classified into which is dependent on Reynolds number Arbitrary cross sections are handled by numerical integration, regular geometries are considerably simpler. Imagine being a single drop of water

Open channel flows Where V = average velocity Rh = hydraulic radius v = kinematic viscosity Arbitrary cross sections are handled by numerical integration, regular geometries are considerably simpler.

Area represents cross sectional area of the fluid Wetted perimeter does not include the free surface These equations are found err where. Just know how to get them.

Rectangular Conduit

Trapezoidal Channel common geometry Engineered (improved) natural channels are reasonably well approximated by trapezoidal equations the geometry is important in drainage engineering

Circular Conduit Sweep angle definition matters, book uses 2a.

Varied Flow Gradually varied flow – change in flow depth moving upstream or downstream is gradual Rapidly varied flow – change in flow depth occurs over a very short distance Ex: waterfall, hydraulic jumps, etc. RVF is outside the scope of this course. Assumptions of velocity on the sides?

Gradually Varied Flow Equation relating slope of water surface, channel slope, and energy slope: Discharge and Section Geometry Gives a relationship for rate oof chNge of depth So and Sf are positive when sloping down in direction of flow y is measured from channel bottom dy/dx =0 means water depth is _______ Variation of Water Surface Elevation Discharge and Section Geometry

Gradually Varied Flow Procedure to find water surface profile is to integrate the depth taper with distance:

yc < y < yn OR yn < y < yn Flow profiles SLOPE DEPTH RELATIONSHIP Steep yn < yc Critical yn = yc Mild yn > yc Horizontal S0 = 0 Adverse S0 < 0 PROFILE TYPE DEPTH RELATIONSHIP Type-1 y > yc AND y > yn Type -2 yc < y < yn OR yn < y < yn Type -3 y < yc AND y < yn

Purpose?

MANning equation (1891) Depth-Discharge Calculator for any open channel implements Manning's equation The equation is the U.S. customary version A drainage engineer in the US should memorize this equation!

Values of Manning n n = f(surface roughness, channel irregularity, stage...) d in ft d = median size of bed material

Summary of open channels All the complications of pipe flow plus ______________________ Importance of Froude Number Fr>1 decrease in E gives increase in y Fr<1 decrease in E gives decrease in y Fr=1 standing waves (also min E given Q) free surface location