## Chapter 3: Fluid dynamics and sediment transport
- Reynolds number = R e = 2rV/ Where r
- Froude number = F r = flow velocity/(acceleration of gravity)(force of inertia) = V/√gD Where D
- F d = F g - F up
- F up = 4/3r 3 f g
- V 2 = 8gr s - f )/3C d f
Velocity
The velocity of a fluid (distance traveled per unit time) determines the type of fluid flow, of which there are two fundamental types: Reynolds number
The transition from laminar to turbulent flow occurs as velocity increases, viscosity decreases, and the roughness of the flow boundary increases, and/or the flow becomes more narrowly confined. These variables were elucidated by the English physicist Sir Osborne Reynolds. The Reynolds number is a mathematical representation of inertial forces/viscous forces.
Reynolds number = R
_{e} = 2rV/Where r is the hydraulic radius, V is the flow velocity, is the fluid density, and is viscocity.
Viscous forces tend to resist fluid motion, keeping flow smooth, while inertial forces generate disordered (turbulent) motions. As such high inertial flows (Re > 5000) tend to be turbulent, and viscous flows (Re < 1500) tend to be laminar. Unconfined fluids moving across open surfaces (windstorms, surface runoff sheet flow, very slow-moving streams, and continental ice sheets) have Re <500-2000 and exhibit laminar flow. Fast-moving streams and turbidity currents have Re >2000. Froude number
The Froude number is a ratio of inertial to gravitational forces for a fluid. It compares the tendency of a moving fluid to continue moving with the gravitational forces that act to stop its motion.
Froude number = F
_{r} = flow velocity/(acceleration of gravity)(force of inertia) = V/√gDWhere D = depth and gD = speed (celerity) of the gravitational wave. F_{r} >1 occurs in fast and/or shallow flows; F_{r} <1 occurs in slow and/or deep flows)
What is a gravity wave?
Throw a stone into a standing body of water and watch the waves move out in concentric paths – this is a gravity wave; now throw a stone into moving water – if you can see the gravity wave move upstream then it is faster than the velocity of the stream. Thus F _{r}<1 otherwise known as tranquil flow, which is typical of most bodies of flowing water. If, however, F_{R}>1 then the velocity of the stream is faster than the gravity wave and rapid flow occurs.
Froude numbers are important to understanding the ripples and other structures that form at the base of rapidly moving streams. Particle motionParticles get picked up when the forces of fluid drag (F_{D}) and fluid lift (F_{L}) work in unison resulting in a net fluid force (F_{F}). Drag exerts a horizontal force, which causes particles to roll, whereas lift raises the particles vertically into the current.
Lift force is an example of Bernoulli’s principle, which states that the sum of the velocity and pressure on an object in a flow must be constant. Whenever a flow speeds up, it exerts less pressure than a slower moving part of the flow.
How do sediments move once they have been lifted?
Sedimentary particles are moved in the bedload by 1) traction (rolling and dragging), or 2) saltation (bouncing, skipping, and jumping). The remainder of particles are carried by suspension.
The relationship between grain size, entrainment, transportation, and deposition is shown in the Hjulstrom diagram, which shows the minimum critical velocity necessary for erosion, transportation, and deposition of clasts of various sizes and cohesion.
Stoke’s LawOnce a particle is entrained in a fluid it begins to sink again under gravitational forces. The distance it travels depends on the drag force of the fluid and the settling velocity of the particle. The settling velocity is calculated using Stoke’s Law, which can be considered as the sum of the gravitational pull downward versus the drag force of the fluid pushing upward.
A freefalling particle will cease to accelerate when the downward force of gravity (F _{g}) is balanced by the frictional force (F_{up}) exerted on the particle by the fluid and the drag forces (F_{d}). The velocity of the particle under these conditions is called the “fall velocity” or “terminal velocity” and can be written as follows:
F
_{d} = F_{g} - F_{up}Stokes calculated the fall velocity for small particles, < 0.1 mm diameter. First, consider the frictional resistance that the fluid offers to movement of a settling sphere F
_{d} = C_{d}r^{2}_{ f} V^{2}/2where F_{d} = resistance (frictional drag), r = particle radius, _{f} is fluid density, and V = settling velocity of sphere.
Then consider the force of gravity pulling the sphere downward F
_{g} = 4/3 r^{3}_{s}gwhere _{s} = density of the sphere and g = acceleration due to gravity
The bouyant force of the liquid is given by F
_{up} = 4/3r^{3}_{f}gwhere _{f } = density of the fluid
Substituting these three factors into the first equation, we get the following: C
_{d}r^{2}_{ f} V^{2}/2 = 4/3 r^{3}_{s}g - 4/3r^{3}_{f}gThis can be simplified into V
^{2} = 8gr_{s} - _{f})/3C_{d}_{ f}If the temperature and fluid density are constant and the sphere and fluid densities known then this equation can be simplified significantly to V = Cr
^{2}where C is a constant given by _{s} - _{f})g/18m. At 20^{o}C, in water, with a sphere density of 2.65 g/cc, C = 3.59 x 10^{4}
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