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Fluid Flow in Rivers

Fluid Flow in Rivers. Outline Flow uniformity and steadiness Newtonian fluids Laminar and turbulent flow Mixing-length concept Turbulent boundary layer Mean boundary shear stress Velocity distribution and the “Law of the Wall” Depth-averaged velocity. Flow in Rivers.

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Fluid Flow in Rivers

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  1. Fluid Flow in Rivers Outline • Flow uniformity and steadiness • Newtonian fluids • Laminar and turbulent flow • Mixing-length concept • Turbulent boundary layer • Mean boundary shear stress • Velocity distribution and the “Law of the Wall” • Depth-averaged velocity

  2. Flow in Rivers • Rivers are nonuniform, unsteady, Newtonian, hydraulically roughturbulent flows

  3. Open Channel Flow (Bridge, 2003)

  4. Open Channel Flow

  5. Fluid Viscosity (Bridge, 2003)

  6. Molecular viscosity is dependent on shear and it has a yield strength (paint) Types of fluids Molecular viscosity is independent of the magnitude of shear once yield strength is exceeded (mud and lava flows) Molecular viscosity is independent of the magnitude of shear (rivers) (Bridge, 2003)

  7. Laminar flow, Re < 2000 (viscous forces dominate) Types Flows Turbulent flow, Re > 4000 (turbulent forces dominate) Large masses of fluid “eddies” are being transported (Hornberger et al., 1998)

  8. Laminar and Turbulent Flow • For 1 m deep flow: Re = 300, U ~ 0.0003 m/s Re = 3000, U ~ 0.003 m/s • Susquehanna River near Waverly, PA d ~ 2 m, U ~ 1 m/s, Re ~ 2,000,000

  9. Turbulent velocity time series (Bridge, 2003)

  10. Turbulent Velocities

  11. Mixing length concept l: the vertical distance over which the momentum of a fluid parcel is changed (mixing length) Reynolds Stress (Bridge, 2003)

  12. Turbulent boundary layer (Bridge, 2003)

  13. Viscous sublayer streaks (Bridge, 2003)

  14. Turbulent “bursting” process ~70% of all turbulence in open channel flows is due to the bursting process (Allen, 1985; Bridge, 2003)

  15. Grain roughness effects on turbulent boundary layer (Julien, 1998)

  16. (Falco, 1977) Large-scale Turbulent Motions (Nakagawa & Nezu, 1981) (Ferguson et al., 1996) (Belanger et al. 2000)

  17. Boundary Shear Stress Conservation of downstream momentum: Impelling force (downstream component of weight of water) = resistive force (Bridge, 2003)

  18. “Law of the Wall” • Derived from Reynolds-averaged Navier-Stokes equations (Gomez, 2006) • Nearly universal use • Important assumptions: • Prandtl’s mixing length theory • Zero slope (?!) • Constant shear stress (no vertical gradient; ?!) neg. 0 0 (Taylor, 1921; Prandtl, 1925, 1926)

  19. Utility of the “Law of the Wall”

  20. RS  LW  DS Reynolds stress Law of the wall Depth-slope product For a mobile upper stage plane bed (mm-scale bedwaves), the boundary is essentially flat. Thus, the mean shear stress determined using a Reynolds stress projection, the law of the wall, and the depth-slope product (relative water surface elevation; WSE) are nearly identical (from Bennett et al., 1998).

  21. Reynolds Stress vs. Velocity Gradient Shear Stress over a Fixed Dune

  22.  y (m)  y4,u4  i=4  y3,u3   u (m/s) Depth Averaged Velocity

  23. Fluid Flow and Stream Restoration • Conforms to physics • Based on steady, uniform flow conditions, there are three (3) methods to determine near-bed shear stress • Distributions of velocity should be predictable

  24. Fluid Flow in Rivers Conclusions • River flow is unsteady, non-uniform, turbulent, and hydraulically rough • River flow can be treated as a boundary layer, and its distribution of velocity can be determined • Turbulence is derived from bursting process • Bed shear stress can be determined from bulk hydraulic parameters and velocity profiles

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