Post on 25-Dec-2015
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
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
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
Understanding Flow and Sediment Transport
• Conservation Relations– Water Mass (aka Continuity)– Momentum (aka Newton’s 2nd Law – F=MA)– Energy
• Constitutive Relations– Flow Resistance (Manning Equation)– Sediment Transport (Shields, Hjulstrom, Bagnold)
Pressure and Shear
• Shear (τ) - exerted to surface ║ Shear (τ) = F/A
• Pressure – exerted to surface = F/A┴
Stress and Strain
= velocity gradient
Shear (τ) = F/A
Shear Stress deforms blockDeformation = StrainStrain proportional to θ
θ
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 θ with time
/du dy
= velocity gradient τ = Shear Stress
Kinematic Viscosity
v
• Viscosity constant at given T; ρ doesn’t depend on type of shearing stress or duration of stress – Newtonian Fluid
• T↑ μ↓• Kinematic viscosity determines extent to which fluid
flow exhibits turbulence
μ = viscosityρ= density
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
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
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.
Shear Stress: Laminar vs Turbulent Flow
du
dy
• Add apparent viscosity or eddy viscosity (η) to turbulent flow shear stress equation
• Turbulence exerts larger shear stress on adjacent fluids than laminar
( )du
dy
Laminar Flow Turbulent Flow
Reynolds Number
• Balance between inertial forces (cause turbulence) and viscous forces (suppress turbulence)
• Laminar: Re < 1000 – viscous dominate; shallow depth or low velocity
• Turbulent: Re >1000 – inertial forces dominate; deep or fast flow
Re = URρ/μ = UR/νU = mean flow velocity ρ = density
R = hydraulic radius (A/P) μ = viscosityν = kinematic viscosity (μ/ρ)
Depth vs Hydraulic Radius
• Some equations use D (or L) – developed in pipes and adopted for open channels
• In wide, shallow channels, R≈D so substitution is ok and simplifies equations
• In deep or incised channels – this is not true and errors are introduced
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
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
Boundary Shear Stress
0 hR S = boundary shear stress= fluid density
= slope (gradient)= hydraulic radius
= hydraulic radius
= cross-sectional area/wetted perimeter
Boundary Shear Stress in Open Channel
• Newton’s 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
Depth-Slope Product
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
Shear Velocity
• Shear stress at bed function of shear velocity (cm/s)• In rivers:
– U* = √gDS 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
UU** = = √√ττoo//ρρU* = Shear VelocityU* = Shear Velocity
ττoo = Boundary Shear Stress = Boundary Shear Stress
ρρ = Fluid Density = Fluid Density
Froude Number
• Ratio between inertial and gravity forces• Gravity influences way fluid transmits shallow water
waves• Dimensionless value (like Re)
r
UF
gL
= Froude Number
= velocity of shallow water wave
= mean flow velocity
g = gravitational acceleration L = water depth
Froude Number
• Fr < 1 Tranquil, Streaming, Subcritical– Velocity of wave > flow velocity
• Fr > 1 Rapid, Shooting, Supercritical– Waves cannot propagate upstream
• Fr has relationship to flow regimes – Defines characteristic bedforms that develop
during flow over a bed
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
U = U = C√R/SC√R/S
Manning Equation
• Similar to Chezy Equation• Manning’s n is presumed to be constant for a given
channel framework• Manning’s 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
Manning’s n
• Can look up n in tables• Can calculate n• Can look up values in a photo guide from
USGS (Barnes, 1968)