CE 3372 WATER SYSTEMS DESIGN Lecture 12 Introduction

CE 3372 WATER SYSTEMS DESIGN Lecture 12: Introduction to Open Channel Hydraulics

OUTLINE • Flow Terminology • Energy Equation • Critical Depth/Flow • Flow Profiles for GVF • Manning’s Equation

OPEN CHANNEL DESIGN CONCEPTS • Interest to engineers: • Water surface elevation (WSE) (minimize impact/reduce floods) • Discharge –Depth relationships • Channel design

OPEN CHANNELS • Conduits whose upper boundary of flow is the liquid surface.

TYPES OF FLOW • Steady Flow – flow, depth and velocity may differ from point to point but remain constant over time Temporal • • 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 Spatial • Non-uniform Flow – velocity is not the same at every point

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 crosssection 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 Datum So 1

OPEN CHANNEL NOMENCLATURE • Slope of the water surface is the slope of the HGL, or slope of WSE (water surface elevation). HGL y z Datum Swse So 1 1

OPEN CHANNEL NOMENCLATURE • Slope of the energy grade line (EGL) is called the energy or friction slope. EGL HGL V 2/2 g Q=VA y z Datum Sf Swse So 1 1 1

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 En erg Hyd rau yg lic g rad el rad e ine The energy grade line (EGL) is: z_head + P_head + v_head The hydraulic grade line (HGL) is at the water surface (EG line L) (HG L) velocity pressure (depth) elevation Sketch of steady flow in a channel Profile grade line is the channel bottom The head loss is depicted as the difference between a horizontal zeroloss energy grade line and the energy grade line

ENERGY RELATIONSHIPS • Energy equation for closed conduits • Energy equation for open conduits

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

CRITICAL DEPTH • Depth of flow for a given discharge, where the specific energy is at a minimum • Occurs when d. E/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

Alternate Depths: y A = By B Plug & chug. Solve for y 3 roots – 1 negative = 2 Q=qy depths where q is the discharge/unit width of

OPEN CHANNEL FLOWS • Open channel flow is also classified by the Froude number • Critical depth, yc occurs at Fr = 1

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-section CRITICAL FLOW T dy Has a minimum at yc y A d. A P More general definition of Fr

CRITICAL FLOW – RECTANGULAR CHANNEL T Ac yc 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

OPEN CHANNEL FLOWS Where V = average velocity Rh = hydraulic radius v = kinematic viscosity

Area represents cross sectional area of the fluid Wetted perimeter does not include the free surface

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 2 a.

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.

GRADUALLY VARIED FLOW • Equation relating slope of water surface, channel slope, and energy slope: Discharge and Section Geometry 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:

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

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 free surface location • 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)

NEXT TIME • Introduction to Storm Water Management Model • Introduction to SWMM software • Workshop for an uncomplicated open channel (bring your computer!)