Fluid Dynamics

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Fluid Dynamics

• Stream Ecosystems

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Fluid Dynamics Lecture Plan

• First consider fluids, stress relationships and
  fluid types
• Then consider factors affecting fluid flow, flow
  velocity, and behavior in pipes vs open channels
• Then understand what controls sediment movement
• Finally put flow and sediment together to
  understand relationships to channel form and
  erosion/deposition in stream systems

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Fluids

• Substances with no strength
• Deform when forces are applied
• Include water and gases
• Body Forces act on whole or bulk of fluid
• Resolve forces within plane of surface of body so
  forces distributed in plane

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Understanding Flow and Sediment Transport

• Ability of river to erode and transport sediment
  represents a balance between driving and
  resisting forces
• Flow and resistance equations are at the heart of
  the discussion

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Understanding Flow and Sediment Transport

• Conservation Relations
• Water Mass (aka Continuity)
• Momentum (aka Newtons 2nd Law FMA)
• Energy
• Constitutive Relations
• Flow Resistance (Manning Equation)
• Sediment Transport (Shields, Hjulstrom, Bagnold)

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Pressure and Shear

• Shear (t) - exertedto surface
• Shear (t) F/A
• Pressure exerted - to surface F/A

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Stress and Strain
Shear (t) F/A
Shear Stress deforms block Deformation
Strain Strain proportional to ?
?
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Viscosity

• Measure of internal friction of fluid particles
• Molecular cohesiveness
• Resistance fluid has to shear (or flow)
• Dynamic viscosity µ shear stress/rate of
  change of ? with time

t Shear Stress
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Kinematic Viscosity
µ viscosity ? density

• Viscosity constant at given T ? doesnt depend
  on type of shearing stress or duration of stress
  Newtonian Fluid
• T? µ?
• Kinematic viscosity determines extent to which
  fluid flow exhibits turbulence

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Types of Fluid Flow

• Laminar Flow flow persists as unidirectional
  movement
• Molecules flow parallel
• Movement up and down by diffusion
• Turbulent Flow highly distorted flow
• Large scale flow perpendicular to direction of
  flow
• Transfer of movement up and down by macroscale
  processes
• Turbulence irregular and random component of
  fluid motion
• Eddies highly turbulent water masses

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Laminar vs Turbulent Flow

• Laminar flow velocity constant at a point over
  time
• Turbulence
• Most flows turbulent
• Slow settling velocity upward motion of water
  particles
• Increases effectiveness of fluid in eroding and
  entraining particles from the bed but less
  efficient transport agent
• Velocity measured at a point over time tends
  towards an average value but varies from instant
  to instant
• Resists distortion to much greater degree than
  laminar flow
• Apparent viscosity eddy viscosity

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Cross-sectional Measurements of Stream Channels

• You will see lots of different variables, terms,
  and ways of expressing channel characteristics
• Need to spend a little time understanding what
  they are so that you can move between and among
  equations and measurements.

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Max Depth(Stage)
Wetted Perimeter
Top Width
Hydraulic Radius A/P
Mean Depth Area/Top Width
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Shear Stress Laminar vs Turbulent Flow
Laminar Flow
Turbulent Flow

• Add apparent viscosity or eddy viscosity (?) to
  turbulent flow shear stress equation
• Turbulence exerts larger shear stress on adjacent
  fluids than laminar

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Reynolds Number
Re UR?/µ UR/?
U mean flow velocity ? density R
hydraulic radius (A/P) µ viscosity ?
kinematic viscosity (µ/?)

• Balance between inertial forces (cause
  turbulence) and viscous forces (suppress
  turbulence)
• Laminar Re lt 1000 viscous dominate shallow
  depth or low velocity
• Turbulent Re gt1000 inertial forces dominate
  deep or fast flow

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Depth vs Hydraulic Radius

• Some equations use D (or L) developed in pipes
  and adopted for open channels
• In wide, shallow channels, RD so substitution is
  ok and simplifies equations
• In deep or incised channels this is not true
  and errors are introduced

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Velocity Profiles and Bed Roughness

• In Turbulent Flow laminar/near laminar flow
  occurs only very near bed
• Smooth beds molecular viscous forces dominate
  in thin layer close to bed boundary
• Viscous sublayer / laminar sublayer
• Rough/Irregular beds
• Coarse sand or gravel
• Viscous sublayer destroyed by particles extending
  through layer
• Obstacles generate eddies at boundary of flow
• Presence/absence of sublayer important factor
  in initiating grain movement

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Boundary Shear Stress

• As fluid flows across bed stress that opposes
  motion of the fluid exists at the bed surface
• Force/unit area parallel to bed
• Extremely important variable in determining
  erosion and transport of sediment on the bed
• F (fluid density, slope of bed, water depth, flow
  velocity)
• Boundary Shear Stress tends to increase as
  velocity increases though in complex ways

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Boundary Shear Stress
cross-sectional area/wetted perimeter
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Boundary Shear Stress in Open Channel
Depth-Slope Product

• Newtons 2nd Law of Momentum
• Calculate boundary shear stress of flow moving
  down channel
• Adds g for gravitational acceleration to account
  for weight of water moving along channel length

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Boundary Shear Stress

• BSS determined by force that flow exerts on bed
  and related to flow velocity determines erosion
  and transport of sediment on bed below a flow
• BSS increases directly with
• ? fluid density
• ? diameter and depth of the stream channel
• ? slope of stream bed
• Greater ability to erode and transport sediment
• Water vs air
• Larger stream channels vs smaller
• Higher gradient streams vs lower

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Shear Velocity
U Shear Velocity
to Boundary Shear Stress
? Fluid Density

• Shear stress at bed function of shear velocity
  (cm/s)
• In rivers
• U vgDS D depth S slope
• Assumes steady, uniform flow
• Average shear velocity of section of channel
• Warning D can be a problem better to use R
• This is still based on flow in pipes

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Froude Number
g gravitational acceleration
L water depth

• Ratio between inertial and gravity forces
• Gravity influences way fluid transmits shallow
  water waves
• Dimensionless value (like Re)

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Froude Number

• Fr lt 1 Tranquil, Streaming, Subcritical
• Velocity of wave gt flow velocity
• Fr gt 1 Rapid, Shooting, Supercritical
• Waves cannot propagate upstream
• Fr has relationship to flow regimes
• Defines characteristic bedforms that develop
  during flow over a bed

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Chezy Equation

• Velocity directly proportional to square root of
  RS product where R A/P S Slope
• Chezy coefficient (C) is a constant of
  proportionality related to resisting factors in
  system
• Equation balances flow velocity with resisting
  forces associated with bed roughness

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Manning Equation

• Similar to Chezy Equation
• Mannings n is presumed to be constant for a
  given channel framework
• Mannings n is also called Manning roughness
  coefficient
• Need estimate of n for each stream reach
• Can be controlled by sediment grain size or
  bedforms controlled by Froude number

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Mannings n

• Can look up n in tables
• Can calculate n
• Can look up values in a photo guide from USGS
  (Barnes, 1968)

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Mannings n Examples
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Mannings n Examples