### Clastic transport and fluid flow

Transport: Weathered rock is transported from source areas to depositional sites by three kinds of processes:

• Direct fluid flows of air, water, and glacial ice
• Wet, gravity-driven mass wasting,
• Dry, gravity-driven mass wasting,

### Fluid-driven Transport:

Hydraulics - the properties of fluid flow

Fluid plays an important role in most models of sediment transport. Thus knowledge of hydraulics, the science of fluid flow, is essential to sedimentation and stratigraphy. Fluids resist forces that tend to change their volume, but readily alter their shape in response to external forces. The ability of a fluid to entrain particles is dependent on:

• Density
• Velocity
• Viscosity (resistance to shearing)
• Grain size
The first two are straightforward.

Density: Mass per unit volume. Typically measured in Kg/m3 or the equivalent g/cm3. Examples:

• seawater = 1.03 g/cm3
• freshwater = 1.0 g/cm3
• glacial ice = 0.9 g/cm3
• air < 0.001 g/cm3

Velocity: Distance traveled per unit time.

Simulation of viscous (below)
and non-viscous (above)
fluid from Wikipedia
Viscosity: the constant of proportionality between shear stress and shear rate. Stated simply, a more viscous fluid resists deformation more strongly than a less viscous. All fluids will deform when stressed. The more they are stressed, the more they are deformed. For most fluids, this relationship is directly proportional and can be written as follows:

τ = μ*du/dy

Where τ is shear stress, du/dy is shear rate, and μ is viscosity. Fluids of this type are called Newtonian fluids. Viscosity can be described as internal friction. Units of this force are in Poise (P) (=1 g * cm-1 * s-1). The names Poiseuille and the shortened Poise are from the French physician, Jean Louis Poiseuille (1799 - 1869).

Examples of the viscosities of common substances at 20 deg. C:

 Water 1.002 x 10-2 Poise Air ~1.7 x 10-5 P Natural gas 1.0 x 10-5 P Glacial ice 1.0 x 1011 P Mantle rock 1.0 x 1020 P

Viscosity is temperature dependent (compare honey that has been heated on a stove with honey coming from a refrigerator.) This table demonstrates the change of water viscosity with temperature:

 Temperature (deg. C) Viscosity cP 0 1.792 20 1.002 40 0.656 60 0.469 80 0.357 100 0.284

### Scaling issue - boundary layers:

When a fluid flows around an object (or an object moves through a fluid) the region directly adjacent to the object - the boundary layer - is dominated by viscous and electrostatic forces that make it tend to adhere to the object. The thickness of this layer varies with:
• Viscosity
• Veocity
The presence of this boundary layer has important consequences for sediment transport.

Laminar and turbulent flow from
Physics and Chemistry for IG and A level
Flow conditions: Although fluid flow is describable by the continuous variables given above, it typically falls into one of two discrete states:
• Laminar flow: Fluid particles move uniformly in subparallel sheets or filaments. Characteristic of slow moving or very viscous material (E.G. glacial ice)

• Turbulent flow: Fluid particles move in random haphazard pattern. Characteristic of high velocity material. High erosive effect (E.G. surface winds)

The image of cigarette smoke (right) shows the abrupt transition between these states.

### Numeric descriptors of flow:

Relationships of velocity, density, viscosity and flow state are numerically described using two dimensionless numbers:
• Reynolds Number
• Froude Number

### Reynolds Number:

Named for British physicist Sir Osborne Reynolds, describes the relative strength of inertial and viscous forces in a moving fluid by giving their dimensionless ratio:

inertial forces/viscous forces

As it happens, this relationship also describes the relationships between the transition from laminar to turbulent flow such that:

• Velocity (higher velocity --> turbulent)
• Viscosity (higher viscosity --> laminar)
• The roughness of the flow boundary (rougher --> turbulent)
• the confinement of the flow (more confined --> turbulent)

Reynolds Number = Re = 2rVρ/μ

Where:

• r is the hydraulic radius ((2A/P) where A = cross-sectional area and P = wetted perimeter)
• V is the flow velocity (m/s)
• ρ is the fluid density (kg/m3)
• μ is viscocity (kg/(m.s)).

Note: The numerator shows inertial forces, or the tendency of discrete particles of fluid to resist changes in velocity and continue to move uniformly in the same direction. The denominator - viscosity, or resistivity to shearing or deformation. The transition from laminar to turbulent flow occurs between Re of 500 and 2000. Typically, we see:

 Environment: Re: Flow: Slow flow and unconfined fluids moving across open surfaces (surface runoff sheet flow, slow streams, continental ice sheets) <500 laminar 500-2000 transitional Rapid constricted flow (Fast streams, turbidity currents) >2000 turbulent

Rain on windshield from BetterPhoto.com
Ultimately, the Reynolds number addresses the behavior of the boundary layer. Consider the familiar example of raindrops on the windshield of a moving car:

• At low speed, raindrops roll downhill on the windshield under the influence of gravity. They can do this because the vehicle is operating at a low Reynolds number so its boundary layer is thick enough completely to encompass the raindrops.
• At higher speeds (and higher Reynolds numbers), the raindrops begin to move upward because the boundary layer has become thin enough that the drops are poking out of it and being blown by the surrounding medium.
• At very high speeds (don't try this at home) the drops move in random directions because the bondary layer is now so thin that the currents of the surrounding medium are channelled through minor irregularities of the windshield's topography (scratches, half-cleaned birdshit stains, etc.)

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< 500) 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.

Why does a sedimentologist care? Because turbulent flow carries much greater erosive force than laminar flow.

Raft caught on standing wave from AbsoluteAstronomy.com

### Froude Number:

Elucidated by British physicist William Froude, a mathematical representation of the ratio between fluid inertial forces and fluid gravitational forces. This describes the tendency of a moving fluid to continue moving despite the gravitational forces that act to stop its motion.

Fr = flow velocity/(acceleration of gravity * force of inertia)

Fr = V/√(gD)

Where:

• V=velocity
• D=depth of flow
• g=gravitational constant.

Consequences:

• Fr < 1 results in tranquil flow. The velocity of gravity waves is greater than the flow velocity (i.e. waves can move upstream).
• Fr > 1 results in rapid flow. The velocity of gravity waves is less than the flow velocity,so no waves propagate upstream.

Gravity waves? 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 Fr <1 otherwise known as tranquil flow, which is typical of most bodies of flowing water. If, however, Fr >1 then the velocity of the stream is faster than the gravity wave and rapid flow occurs.

Why does a sedimentologist care? Because Froude numbers are important to understanding the ripples and other sedimentary structures that form at the base of rapidly moving streams.

### Particle Transport

Particle motion in flowing fluid: We care about hydraulics because of flowing fluids' ability to transport sediment. Why, however, does this actually happen?

Whereas ions in solution are simply part of the fluid, clasts must be moved by mechanical forces. This involves three distinct processes:

• Entrainment: Particles picked up into the transport medium (water, air, etc.)
• Transport: Movement within the transport medium
• Deposition: Settling out of the transport medium

• Entrainment: Clasts are initially mobilized by two forces:
• fluid drag (FD)
• fluid lift (FL)

Drag exerts a horizontal force, which causes particles to roll in the direction of current flow, whereas lift raises the particles vertically into the current (obliquely to 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. We are familiar with this from the airfoil shape of aircraft wings, deliberately engineered to create lift (right above). Fluid moving over a clast also generates lift (right below).

In flowing streams, FD and FL combine to yield a composite fluid force, FF.

The sediment, itself, is transported as three distinct loads:
• Dissolved load: The fraction in solution as ions.

• Suspended load: That fine portion that is kept in constant suspension by electrostatic and viscous interactions with the surrounding water. Generally clay or silt sized particles.

• Bed load: That portion that cannot be kept in constant suspension. Generally sand sized and larger. The bed load moves in two manners:
• saltation: bouncing along the stream bed, or being repeatedly picked up and put down by FF.
• Traction: The remainder that rolls or slides in constant contact with the bed.

There is a relationship between a stream's velocity (energy) and the size clast that it can transport. Intuitively we sense that larger clasts need more energy to be moved than smaller ones, but reality is more subtle. This is shown in the Hjulstrom diagram, that gives a zone of sediment transport in black. The upper limit is the velocity at which a clast of a given size is entrained, the lower limit is the velocity at which an entrained clast is deposited.
• No surprise that the smallest grains are deposited at the lowest flow velocities.
• But why are sand sized particles entrained before clay particles? (See if you can figure it out.)

Above all, larger clasts are entrained at higher velocities. Why?

### Stokes' Law:

Developed by Sir George Gabriel Stokes, An object falling through a fluid experiences three forces:
• The force of gravity (Fg), accelerating it downward
• Bouyancy (Fup)
• Drag (Fd), pushing it upward.

Terminal fall velocity: The speed with which it falls is the balance of these forces. Gravity tends to accelerate it downward, however, the laws of fluid dynamics tell us that the faster the object falls, the more drag it experiences, slowing it down. Eventually it reaches a settling rate, a point at which drag and bouyancy balance gravitational acceleration, the terminal fall velocity (Fd).

Essentially, this is expressed as Fd = Fg - Fup

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:

• Fd = Cdπ(d2/4)(ρf V2/2)

where:

• Fd = resistance (frictional drag)
• Cd = Cd=drag coefficient (constant)
• d = particle diameter
• ρf is fluid density
• V = settling velocity of sphere.

Then consider the force of gravity pulling the sphere downward:

• Fg = 4/3 π(d/2)3ρsg

where ρs = density of the sphere and g = acceleration due to gravity.

The bouyant force of the liquid is given by:

• Fup = 4/3π(d/2)3ρfg

where ρf = density of the fluid

Substituting these details into the original equation gives us:

• Cdπ(d2/4)(ρf V2/2) = 4/3 π(d/2)3ρsg - 4/3π(d/2)3ρfg

This can be simplified into:

• V2 = 4gd(ρs - ρf)/3Cdρf

If the temperature and fluid density are constant and the sphere and fluid densities known then this equation can be simplified significantly using the Reynolds number relationship to:

• V = Cd2

where C is a constant given by (ρs - ρf)g/18μ. At 20 deg C, in water, with a sphere density of 2.65 g/cc, C = 3.59 x 104

The punchline: In words, with density and viscosity constant, velocity increases as the square of grain diameter.

If we rotate the arrangement 90 degrees: With density and viscosity constant, the velocity required to move a clast through drag increases as the square of grain diameter.