Hydraulics for Hydrographers Basic Hydrodynamics AQUARIUS Time-Series Software™ Aquatic Informatics Inc.

Derivation of a rating equation Froude Number Preview Properties of Water States of flow Forces acting on Flow Derivation of a rating equation Froude Number

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.

Water Flow is Governed by Gravity and Friction Gravity – relates to Specific Weight Specific Weight is Weight/Volume = γ = ρg γ = ρg ; ρ = density ; g = gravity γ = 98100 N/m3 for water Friction – relates to Viscosity and surface area ViscosityWater = 0.3 to 1.6 Specific Weight is an important concept in hydraulics. It allows us to equate a volume of water (m3) to a force (N) knowing the density of water. This is one of the most important forces for generating flow. Note that over the range temperature of most surface water streams the specific weight of pure water is relatively constant. Other factors - such as suspended sediment load - can be more important variables affecting specific weight in natural streams. Viscosity is a force that resists the generation of flow. A highly viscous substance will not generate much velocity even if the specific weight of the substance is high and vice versa. Viscosity of pure water is variable with respect to temperature within the range of temperatures normally encountered but to put this in perspective, water 100,000 times more fluid than ketchup, 10,000 times more fluid than honey, 1,000 time more fluid that castor oil, 100 time more fluid than olive oil, and 5 times more fluid than blood.

When uniform, flow lines are parallel Velocity and depth do not vary over distance

Velocity and depth do not vary over time Steady Flow Velocity and depth do not vary over time If depth and/or discharge fluctuate, then flow is unsteady The steady flow assumption allows us to ignore local acceleration of water in the stream greatly simplifying the equations governing flow.

Assumptions of steady and uniform flow depend on scale: 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. Outside of a laboratory, flow is rarely steady and uniform over very great distances or for very long periods of time. Gradually varied flow is a concept that allows us to conceptualize a reach of river over a period of time that is sufficiently uniform and steady to be useful for approximation. You can think of the river passing from one steady uniform state to a different steady uniform state sufficiently gradually that at any instant of time our assumption is practically true.

Discharge must remain constant along a channel Q=A x V Uniform Flow Discharge must remain constant along a channel Q=A x V Water cannot be created or destroyed. Therefore – for a uniform channel - the product of width depth and velocity is constant along the length of the streamtube.

Pressure is linearly related to flow depth When we measure stage we are usually measuring a pressure and converting the pressure to a length by virtue of a linear relationship…. Pressure

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) Pressure Potential energy is the energy available to create velocity from a static pool with zero approach velocity. Pressure Head is the dominant form of energy during low flow and in low-gradient streams.

The energy an object possesses because of its motion Kinetic Energy The energy an object possesses because of its motion For Fluids Kinetic Energy per unit weight = Also called “Velocity Head” Kinetic energy is the energy that is translated through the stream reach from upstream. Kinetic energy is often the dominant form of energy during high flow situations and in steep gradient streams.

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 Knowing the Force (from the specific weight of water) we can calculate pressure at any depth.

The Bernoulli Equation p = pressure g = specific weight z = height above a datum v = velocity g = acceleration of gravity This important thing to notice about this equation is that if we know something about Total Head and the properties of water then we can say something about velocity. This equation is fundamental to the use of stage to estimate discharge.

Conservation of Momentum A mass keeps a constant velocity unless subjected to a force (Newton’s 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 Conservation of momentum allows us to make predictions about the forces resisting the sum of the potential and kinetic energy in a stream reach. Understanding the energy in the system is only part of the story. We now need to know something about the resistance to flow….

Friction Head Loss where Hf=Head loss due to friction, K is a constant, ν = 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

Chezy rearranged the equations for force to solve for velocity. Chezy’s 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. 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

Manning’s contribution is that: Manning’s Equation Chezy’s ‘C’ varies with stage, which limits the usefulness of the Chezy equation. Frictional resistance is not a constant but varies with respect to mass. Manning’s contribution is that: Which gives:

Derivation of the Stage-Discharge Equation The stage discharge equation can be derived from the Manning equation by first multiplying Velocity times Area: However, we don’t 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)…. 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:

The Stage-Discharge equation ‘β’ 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 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 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 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…

Derivation using Velocity Head 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 a 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.

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?

Ec Specific Energy Critical Depth 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 ½ of Depth.

Dimensionless number comparing inertial (V) and gravitational forces. Froude Number Dimensionless number comparing inertial (V) and gravitational forces. Where, v = Velocity; g = gravitational acceleration; and D = Depth Sub-critical < 1; critical = 1; super-critical >1

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

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