Fluid Dynamics

Stream Ecosystems

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.

Max Depth(Stage)

Top Width

Hydraulic Radius = A/PMean Depth = Area/Top Width

Wetted Perimeter

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