Hydraulics for Hydrographers Basic Hydrodynamics AQUARIUS Time-Series Software Aquatic Informatics Inc. Preview Properties of Water States of flow Forces acting on Flow Derivation of a rating equation Froude Number

www.aquaticinformatics.com | 2 Understanding River Flow The unique properties of water and some basic physics allow us to make predictions. We will review the concepts that can help us.

www.aquaticinformatics.com | 3 Water Flow is Governed by Gravity and Friction Water Flow is Governed by Gravity and Friction Gravity relates to Specific Weight Specific Weight is Weight/Volume = = g g = g g ; g = density ; g = gravity

= 98100 N/m3 for water Friction relates to Viscosity and surface area ViscosityWater = 0.3 to 1.6 www.aquaticinformatics.com | 4 Uniform Flow When uniform, flow lines are parallel Velocity and depth do not vary over distance www.aquaticinformatics.com | 5

Steady Flow Velocity and depth do not vary over time If depth and/or discharge fluctuate, then flow is unsteady www.aquaticinformatics.com | 6 Steady Flow Assumptions of steady and uniform flow

depend on scale: Gradually Varied Flow allows us to assume steady and uniform flow on a short scale. Rapidly Varied Flow does not permit to approximate flow as steady and uniform. Uniform Flow Discharge must remain constant along a channel Q=A x V Qo Qi S dt

Q A1V1 A2 V2 A3 V3 A4 V4 .... An Vn www.aquaticinformatics.com | 8 Depth Pressure is linearly related to flow depth Pressure www.aquaticinformatics.com | 9 Potential Energy Gravitational Potential Energy

Energy of water above a datum, e.g. sea level Also called Potential Head Pressure Potential Energy Energy of water above the channel bed Also called Pressure Head (=pressure/specific weight) Kinetic Energy The energy an object possesses because of its motion M V 2

E 2 2 For Fluids V Kinetic Energy per unit weight = 2 g Also called Velocity Head Pressure is related to Force Pressure is a force over an area applied by

an object in a direction perpendicular to the surface. Pressure = Force/Area Pressure is conjugate with Volume However, water is incompressible hence: Force/Area = Weight / Area = (Specific weight x Volume)/Area = Specific Weight x (w x l x d)/ (w x l) = Specific Weight x Depth The Bernoulli Equation 2

v z TotalHead TotalEnerg y 2g p = pressure g = specific weight z = height above a datum v = velocity g = acceleration of gravity Conservation of Momentum

A mass keeps a constant velocity unless subjected to a force (Newtons 3rd Law) Streamflow does not accelerate indefinitely However, water is incompressible hence: Gravity and pressure is counteracted by friction Potential and kinetic energy transforms into heat When flow is Steady and Uniform the forces in the Bernoulli equation are exactly balanced by forces resisting flow Friction Head Loss 2

H f K v P L where Hf=Head loss due to friction, K is a constant, = velocity, P= Wetted Perimeter, and L = Length of the channel. K includes channel rugosity; sinuosity; size; shape; obstructions; as well as the density and kinematic viscosity of the water Chezys equation Chezy rearranged the equations for force to solve for velocity.

He simplified the physics by lumping all of the variables that are nearly constant into a constant. 0.5 V CR S 0.5 Where V= velocity, C = a constant; R = Hydraulic Radius; and S = slope The simplifying assumptions

The constant C includes gravitational acceleration and Head loss due to friction, which are assumed to be nearly constant. The Hydraulic Radius is used as an index of both cross sectional area (a component of the specific weight driving flow) and of wetted perimeter (a component of Head Loss due to friction). Slope is used to convert the downward gravitational force to a longitudinal force along the channel Mannings Equation

Chezys C varies with stage, which limits the usefulness of the Chezy equation. Frictional resistance is not a constant but varies with respect to mass. 1 Mannings contribution is that: 1 Which gives: 1 0.67 0.5 V R S n

C R6 n Derivation of the Stage-Discharge Equation The stage discharge equation can be derived from the Manning equation by first multiplying Velocity times Area: 1 0.67 0.5 Q R S A n

However, we dont know n,; and R, S and A are all relatively difficult to monitor continuously Some simplifying assumptions That flow is Pressure Head dominated That n does not vary as a function of stage That S does not vary as a function of stage That R does vary as a function of stage (f1) That A does vary as a function of stage (f2). 0.5 S

0.67 1 Q f1 (H) f 2 (H) n If only we could combine the equations that solve for Radius and for Area into one function Area and Radius as a function of stage Assume that Area and Radius are both linear functions of stage These relations converge where Area = 0 because

R = A/P; call this point PZH then: A = m1(H-PZH) and R = m2(H-PZH) and: S 0.5 Q m 1 m 2 (H - PZH) n 1 0.67 (H - PZH ) 1 .67 The Stage-Discharge equation

Q (H - PZH) a b contains all information about slope (S); roughness (n); river size (m1); channel complexity (m2); physical properties of water; and the Velocity Head component of flow. PZH is the point of convergence for two different linear functions of stage (H R), (H A) that convert stage to a measure of Head The exponent (a) is the exponent of Area as a function of Head (e.g. 1 for a vertical banks, 2 for a banks sloped at a 45o angle); (b) is the exponent of pressure as function of Head (0.67)

The Stage-Discharge equation Q (H - PZH) Where the Point of Zero Head may be equal to the Point of Zero Flow (PZF) for smooth bottom sections. PZH may differ from PZF for irregular channel control sections because the bottom range of stage does not contribute equally to the Specific Weight of water in the water column.

Derivation using Velocity Head v 2 gh For the previous derivation, we assumed that flow is Pressure Head dominated However, even if the flow has significant component of velocity head we can still derive the stage discharge relation From the Bernoulli equation, we can solve for velocity as a function of gravity and Head Derivation using Velocity Head

q 2 gh w h knowing velocity we can solve for discharge by multiplying by width and depth We can rearrange this into the familiar form of the stage discharge relation by combining width with the square root of 2 times gravity thus 1.5

q 2 g w h Derivation using Velocity Head Q H PZH the coefficient B contains information about: the width of the section; gravitational acceleration; assumptions about the physical properties of water; and the Pressure Head component of flow The exponent contains information about the shape of

the stream banks vertical banks would resolve to an exponent of 1.5 and banks sloping back at a 45o angle would resolve to an exponent of 2.5. Specific Energy Energy per unit mass of water at any section of a channel measured with respect to the channel bottom. 2 V E D

2g Specific Energy Tranquil Flow Gentle gradient Critical Flow Turbulent Flow Steep gradient Understanding River Flow

Can you see the velocity head and pressure head components of flow around Trevor? www.aquaticinformatics.com | 29 Specific Energy 2 V

2g Critical Depth Dc 2 Vc gDc This means that we can calculate velocity directly from depth observations

Ec When Flow is critical, say at a sharp break in the channel slope, Velocity Head is of Depth. Froude Number Dimensionless number comparing inertial (V) and gravitational forces. Fr

v gD Where, v = Velocity; g = gravitational acceleration; and D = Depth Sub-critical < 1; critical = 1; super-critical >1 Froude Number The Froude number of a stream can be guessed at by observation If you throw a stone in the stream, the Froude number

is less than unity if ripples can propagate upstream; this means that the flow is Pressure Head dominated. If there is turbulent flow, then the Froude number is greater than unity; this means that the flow is Velocity Head dominated If the flow is passing over a sharp crest, or through a significant channel narrowing, then the Froude number at that point is equal to unity; this means that the flow is critical. Recommended, on-line, self-guided, learning resources USGS GRSAT training

http://wwwrcamnl.wr.usgs.gov/sws/SWTraining/Index.htm World Hydrological Cycle Observing System (WHYCOS) training material http://www.whycos.org/rubrique.php3?id_rubrique=65#hydrom University of Idaho http://www.agls.uidaho.edu/bae450/lessons.htm Humboldt College http://gallatin.humboldt.edu/~brad/nws/lesson1.html Comet Training need to register no cost http://www.meted.ucar.edu/hydro/basic/Routing/print_version/05-stage_discharge.

htm#11 Thank you from the AI Team We hope that you enjoy AQUARIUS! www.aquaticinformatics.com | 34