Basic Hydraulics: Open Channel Flow – I

Open Channel Definitions Open channels are conduits whose upper boundary of flow is the liquid surface Canals, rivers, streams, bayous, drainage ditches are common examples of open channels. Storm and sanitary sewers are are also open channels unless they become surcharged (and thus behave like pressurized systems).

Open Channel Nomenclature Elevation (profile) of some open channel

Open Channel Nomenclature Flow profile related variables Flow depth (pressure head) Velocity head Elevation head Channel slope Water slope (Hydraulic grade line) Energy (Friction) slope (Energy grade line)

Open Channel Nomenclature Cross Sections Natural Cross Section Engineered Cross Section

Open Channel Nomenclature Cross Section Geometry and Measures Flow area (all the “blue”) Wetted perimeter Topwidth Flow depth Thalweg (path along bottom of channel)

Open Channel Steady Flow For any discharge (Q) the flow at any section is described by: Flow depth Mean section velocity Flow area (from the cross section geometry) Depth-area, depth-topwidth, depth-perimeter are used to estimate changes in depth (or flow) as one moves from section to section

Open Channel Steady Flow Types The flow-depth, depth-area, etc. relationships are non-unique, flow “type” is relevant Uniform (normal) Sub-critical Critical Super-critical

Cross Section Geometry Normal, Critical, Sub-, Super-Critical flow all depend on channel geometry. Engineered cross sections almost exclusively use just a handful of convenient geometry (rectangular, trapezoidal, triangular, and circular). Natural cross sections are handled in similar fashion as engineered, except numerical integration is used for the depth-area, topwidth-area, and perimeter-area computations.

Cross Section Geometry Rectangular Channel Depth-Area Depth-Topwidth Depth-Perimeter

Cross Section Geometry Trapezoidal Channel Depth-Area Depth-Topwidth Depth-Perimeter

Cross Section Geometry Triangular Channels Special cases of trapezoidal channel V-shape; set B=0 J-shape; set B=0, use ½ area, topwidth, and perimeter

Cross Section Geometry Circular Channel (Conduit with Free-Surface) Contact Angle: Depth-Area: Depth-Topwidth: Depth-Perimeter:

Cross Section Geometry Irregular Cross Section Use tabulations for the hydraulic calculations

Cross Section Geometry Irregular Cross Section – Depth-Area Depth A4 A3 A2 A1 Area A1 A1+A2 A1+A2+A3

Cross Section Geometry Irregular Cross Section – Depth-Area Depth T4 T3 T2 T1 T1 Topwidth T1+T2 T1+T2+T3

Cross Section Geometry Irregular Cross Section – Depth-Perimeter Depth P4 P3 P2 P1 Perimeter P1+P2 P1 P1+P2+P3

Flow Direction/ Cross Section Geometry Convention is to express station along a section with respect to “looking downstream” Left bank is left side of stream looking downstream (into the diagram) Right bank is right side of stream looking downstream (into the diagram) Left Bank Right Bank Flow Direction

Energy Equation in Open Channel Flow α = velocity head correction factor

Energy Equation in Open Channel Flow When velocity is nearly uniform across the channel the correction factor is usually treated as unity (α = 1) Hence, the energy equation is typically written as

Potential Energy In pressurized systems the potential energy is the sum of the pressure and elevation head. In open channels, elevation is taken at the bottom of the channel, the analog to pressure is the flow depth. Thus the potential energy is the sum of elevation and flow depth

Kinetic Energy Kinetic energy is the energy of motion; in pressurized as well as open channel systems, this energy is represented by the velocity head The sum of these two “energy” components is the total dynamic head (usually just “total head”)

Hydraulic Grade Line The hydraulic grade line is coincident with the water surface. It represents the static head at any point along the channel.

Specific Energy Total energy is the sum of potential and kinetic components: Energy relative to the bottom of the channel is called the specific energy (at a section)

Specific Energy Relationship of Total and Specific Energy (at two different sections) y2 y1

Specific Energy Calculations For example Rectangular channel, Q=100cfs; B=10 ft So Table shows values. Plot on next page = specific energy diagram

Specific Energy Diagram

Critical Depth Specific energy relationship has a minimum point Flow at specified discharge cannot exist below minimum specific energy value Depth associated with minimum energy is called “critical depth” Critical depth (if it occurs) is a “control section” in a channel What is the value of critical depth for the case shown in the previous diagram (and table)?

Flow Classification by Critical Depth Subcritical flow –Water depth is above critical depth (velocity is less than the velocity at critical depth) Supercritical flow – Water depth is below critical depth (velocity is greater than the velocity at critical depth) Critical flow – Water depth is equal to critical depth. 1 to 1 depth-discharge at critical (dashed line) Q3 Q2 Q1

Flow Classification by Critical Depth Classification important in water surface profile (HGL) estimation and discharge measurement. Water can exist at two depths except at critical depth Critical depth important in measuring discharge Sub- and Super-Critical classification determine if the controlling section is upstream or downstream. Sub- and Super-Critical classification determines if computed HGL will be a front-water or back-water curve.

Flow profiles in Long Channels Flow is subcritical. Flow is at critical depth. supercritical.

Conveyance The cross sectional properties can be grouped into a single term called conveyance Manning’s equation becomes Units of conveyance are CFS

Normal depth Normal depth is another flow condition where the slope of the energy grade line, channel bottom, and the slope of the hydraulic grade line are all the same Manning’s equation assuming normal depth is

Normal depth To use need depth-area, depth-perimeter information from channel geometry. Then can rearrange (if desired) to express normal depth in terms of discharge, and geometry. Computationally more convenient to use a root-finding tool (i.e. Excel Goal Seek/Solver) than to work the algebra because of the exponentation of the geometric variables.

Normal depth For example, TxDOT HDM Eq 10-1 is one such Manning’s equation, rearranged to return normal depth in a triangular section (J-shape) where Q = design flow (cfs); n = Manning’s roughness coefficient ; Sx = pavement cross slope; S = friction slope; d = normal depth (ft).