Hydraulics for HydrographersBasic Hydrodynamics

AQUARIUS Time-Series Software™Aquatic Informatics Inc.

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Properties of WaterStates of flowForces acting on FlowDerivation of a rating equationFroude Number

Preview

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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 byGravity and Friction

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Water Flow is Governed byGravity 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

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Uniform Flow

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

Steady Flow

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Velocity and depth do not vary over timeIf depth and/or discharge fluctuate, then flow is unsteady

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.

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Uniform Flow

Discharge must remain constant along a channelQ=A x V

Q A1V 1 A2 V 2 A3V 3 A4 V 4 .... AnV n

Qo Qi S dt

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Pressure is linearly related to flow depth

Pressure

Dep

th

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

For Fluids• Kinetic Energy per unit weight =• Also called “Velocity Head”

2

2VME

g

V

2

2

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

p = pressure g = specific weight z = height above a datum v = velocity g = acceleration of gravity

yTotalEnergTotalHeadg

vz

2

2

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

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

LPvKH f 2

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

5.00.5CRV S

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 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:

6

1

R n

1 C

0.50.67 S R n

1 V

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…

A S R n

1 Q 0.50.67

Some simplifying assumptions

That flow is Pressure Head dominatedThat ‘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…

1267.0

1

0.5

(H)f (H)f n

S Q

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:

67.10.67 121

0.5

)PZH-(H PZH)-(H m m n

S Q

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)

baPZH) - (H Q

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.

PZH) - (H Q

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…

ghv 2

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…

hwghq 2

5.12 hwgq

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.

PZHHQ

Specific Energy

Energy per unit mass of water at any section of a channel measured with respect to the channel bottom.

g

VDE

2

2

Specific Energy

Tranquil Flow

Critical Flow

Turbulent Flow

Gentle gradient

Steep gradient

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Understanding River Flow

Can you ‘see’ the velocity head and pressure head components of flow around Trevor?

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2cD

gV

cc gDV Critical Depth

Ec When Flow is critical, say at a sharp break in the channel slope, Velocity Head is ½ of Depth.

Specific Energy

This means that we can calculate velocity directly from depth observations

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

gD

vFr

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 dominatedIf 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 traininghttp://wwwrcamnl.wr.usgs.gov/sws/SWTraining/Index.htm

World Hydrological Cycle Observing System (WHYCOS) training materialhttp://www.whycos.org/rubrique.php3?id_rubrique=65#hydrom

University of Idahohttp://www.agls.uidaho.edu/bae450/lessons.htm

Humboldt Collegehttp://gallatin.humboldt.edu/~brad/nws/lesson1.html

Comet Training – need to register – no costhttp://www.meted.ucar.edu/hydro/basic/Routing/print_version/05-stage_discharge.htm#11

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Thank you from the AI TeamWe hope that you enjoy AQUARIUS!