Transcript hjkhj - Aquatic Informatics

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
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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.
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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 = 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 parallel
Velocity and depth do not vary over distance
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Steady Flow
Velocity and depth do
not vary over time
If depth and/or
discharge fluctuate,
then flow is unsteady
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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  A1 V1  A2  V2  A3  V3  A4  V4  .... An  Vn
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Depth
Pressure is linearly related to flow depth
Pressure
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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

v
z
 TotalHead  TotalEnergy

2g
2
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 (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
H f  K v  P L
2
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’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.
V  CR S
0.5
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
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.
1
Manning’s contribution is that:
1
C
Which gives:
1 0.67 0.5
V R S
n
n
R6
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 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)….
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:
S0.5
Q
 m1  m 2  (H - PZH) 1  0.67   (H - PZH )1.67
n
The Stage-Discharge equation
Q    (H - PZH)
a b
‘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
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…
q


2g  w  h
1.5
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?
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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 ½ 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/05stage_discharge.htm#11
Thank you from the AI Team
We hope that you enjoy AQUARIUS!
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