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 2 nd 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 block Deformation = Strain Strain 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 = velocity gradient τ = Shear Stress

Kinematic Viscosity 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 = viscosity ρ= 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/P Mean Depth = Area/Top Width Wetted Perimeter

Shear Stress: Laminar vs Turbulent Flow Add apparent viscosity or eddy viscosity (η) to turbulent flow shear stress equation Turbulence exerts larger shear stress on adjacent fluids than laminar Laminar FlowTurbulent 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 R e = URρ/μ = UR/ν U = mean flow velocityρ = density R = hydraulic radius (A/P)μ = viscosity ν = 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 = boundary shear stress = fluid density = slope (gradient)= hydraulic radius = cross-sectional area/wetted perimeter

Boundary Shear Stress in Open Channel Newton’s 2 nd 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 * = √gDSD= depthS= 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 U * = √τ o /ρ U* = Shear Velocity τ o = Boundary Shear Stress ρ = Fluid Density

Froude Number Ratio between inertial and gravity forces Gravity influences way fluid transmits shallow water waves Dimensionless value (like Re) = Froude Number = velocity of shallow water wave = mean flow velocity g = gravitational acceleration L = water depth

Froude Number Fr < 1Tranquil, Streaming, Subcritical – Velocity of wave > flow velocity Fr > 1Rapid, 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 = C√R/S