2.2: Fluid Mechanics

Water, air, ice, and gravity are agents of sediment transport (erosion). Although they have very different physical properties, the first three are all fluids because they flow when shear stress is applied to them. As these materials flow, generally in response to gravity, they can transport sediment. Sediment can also move in direct response to gravity without the aid of a fluid – If you don't believe that, just roll a rock down a hill or push it off the edge of a table! Although the fluid is not needed, its presence can facilitate movement by decreasing friction along existing planes of weakness.

Density and Viscosity

Density and viscosity are the properties of a fluid that most profoundly influence the way in which it flows and its capacity for sediment transport. Density is a measure of mass per unit volume (ex: g/cm³). Liquids (water) have a much higher density than gasses (air) and most naturally occurring solids (minerals) have a higher density than liquid water. One noteworthy exception is water ice, this solid form of water floats because it has a lower density than the liquid form. Viscosity is the measure of a substance’s internal resistance to flow (commonly measured as kg/(m·s) or m²/s). Fluids with a high viscosity have a high resistance to flow; those with a low viscosity have a low resistance to flow. Honey has a much higher viscosity than liquid water, which in turn has a much higher viscosity than air. The viscosity of a given material can also change with temperature; water ice has a much higher viscosity than liquid water and syrup stored in a refrigerator has a higher viscosity than syrup that has been warmed.

Laminar vs. Turbulent Flow

Fluid flow occurs in two very different modes (laminar and turbulent flow) depending primarily on the flow velocity, fluid viscosity, and flow depth. The differences in flow conditions are best considered in terms of flow lines, which represent the paths of individual particles and can be visualized by adding dye tracer from a point source within the water column. Laminar flow occurs where flow lines are broadly parallel to one another, curve smoothly around obstructions and do not form eddies behind them, and stay at the same orientation through time. Laminar flow can be created when flow smoothly streams out of a faucet or where a smooth body moves through shallow water. Turbulent flow exists where flow lines cross and become indistinct, form eddies behind obstructions, or show random changes in direction at a given location through time. Turbulent flow can be generated by increasing flow out of a faucet until it becomes chaotic and contains air.

Reynolds Number is a dimensionless number that quantifies the balance between viscous forces (tendency for particles to smoothly shear past one another) and inertial forces (tendency for moving particles to resist changes in velocity and direction). Laminar flow exists where viscous forces dominate and the fluid moves in an organized fashion; turbulent flow exists where inertial forces and chaotic motion dominate. Reynolds Number can be calculated by the following equation:

where:

V = flow velocity

D = flow depth

ρ = density

µ = viscosity

In nature, the transition between laminar and turbulent flow happens at Reynolds Number values of 500 to 2000. Values of less than 500 are typically laminar and represent organized flow with relatively little capacity for sediment transport. Values above 2000 are typically turbulent and are much more effective at transporting sediment. This equation shows that increasing flow velocity, depth, or density (or decreasing viscosity) will tend to push the flow toward turbulent conditions and increase its potential for sediment transport.

Figure \(\PageIndex{1}\): Examples of laminar and turbulent flow. A) Photograph showing the transition from laminar to turbulent flow in a plume of air rising from a candle. The plume starts of as laminar but transitions to turbulent in the top 1/3 of the frame (Gary Settles via Wikimedia Commons, CC BY-SA 3.0). B) Water flowing over the left side of the dam starts of as laminar and transitons to turbulent flow downstream. Water flowing over the right side of the dam almost immediately transitions to turbulent flow (Tangopaso via Wikimedia Commons; public domain).

 

Video \(\PageIndex{1}\): Why laminar flow is awesome! Great flume- and field-based examples of laminar and turbulent flow.

 

Settling Velocity

Stoke’s Law is an equation that shows the relationship between settling velocity, fluid density, particle density, and particle size. This equation is used for small particles (<0.1 mm diameter) where it is assumed that the particles are spherical and that the water moving around them is experiencing laminar flow. Given these assumptions, settling velocity can be calculated as follows:

where:

V is the settling velocity

ρ_s is the density of the particle

ρ_f is the density of the liquid

g is the gravitational constant

D is the diameter of the sphere

μ is the viscosity of the liquid

Effectively, this equation tells us that, for small particles, the most important factors are the diameter of the particle and the difference in density between the particle and the fluid. It also tells us that increasing the viscosity of the fluid decreases the settling velocity.

Video \(\PageIndex{2}\): A plain language explanation of Stoke's Law.

For particles larger than very fine sand, turbulent flow around the particle is more likely and the viscosity of the fluid becomes less important. In this case, its best to use a modified version of the equation:

where:

V is the settling velocity

R is the specific gravity of the particle

D is the diameter of the particle

μ is the viscosity of the liquid

g is the gravitational constant

This equation is very similar, the big take away is that the extra parts in the denominator factor in the drag that comes with larger particles (by increasing the value of the denominator you decrease the settling velocity). Taken together, these equations tell us several basic, but important, things about how particles behave when settling out of the water column:

 

Size

Bigger particles settle out faster than smaller ones (if all other variables are held constant). Some representative settling velocities for different sizes of spherical quartz particles are:

• 0.001 mm (clay) = 0.0001 cm/s
• 0.01 mm (silt) = 0.01 cm/s
• 1 mm (sand) = 15 cm/s
• 10 mm (pebble) = 71 cm/s

The relationship between particle size and settling velocity explains why the largest clasts are found at the bottom of graded beds and why many sedimentary rocks are at least moderately sorted.

Density

Particles with a high density settle out faster than those with a low density. The relationship between density and settling velocity explains why, in beach sands, relatively small grains of dark magnetite (density = 5.21 g/cm³) commonly occur with much larger quartz grains (density = 2.65 g/cm³).

Shape

It is also worth noting that particle shape can profoundly influence settling velocity and that these differences in shape are not factored in to the simplified equations provided above. Many sandstones contain broadly spherical sand grains of a certain size fraction mixed with considerably larger sheet-like grains of mica. The flat shape of the mica grains greatly increases the drag force which slows the descent of the micas. The overall result is that, from a hydrodynamic perspective, large platy grains settle out at a velocity similar to smaller, spherical grains.

Froude Number: Supercritical vs. subcritical flow

Perhaps the most abstract fluid flow concept is that of supercritical and subcritical flow – conditions that can be quantified using the Froude Number:

Where: F_r is Froude Number (dimensionless)

V is flow velocity

g is the gravitational constant

L is flow depth

This number quantifies the relationship between inertial forces (tendency for moving particles to resist changes in velocity and direction) and gravitational forces (downward attraction toward the Earth). In terms of what this means to flowing water, calculation of Froude Number reveals whether a wave (generated by a disturbance or obstruction) is capable of moving upstream. A Froude number of less than one indicates tranquil, subcritical flow where the wave could move upstream and influence material upstream of it (wave velocity > current velocity). A Froude Number of greater than one indicates supercritical flow where wave energy cannot move upstream and cannot exert an upstream influence (current velocity > wave velocity).

A disturbance in supercritical flow (F_r>1) is analogous to sounding an airhorn out the window of a supersonic jet; the sound waves will immediately fall behind and will never get ahead of the jet. A more sedimentological example is to think about throwing a pebble into a river; if the ripples can move upstream the flow is tranquil (F_r < 1), if they cannot, the flow is supercritical (F_r > 1). Standing waves in a rapidly moving current form when the flow is transitional between these two states (F_r = 1).

Video \(\PageIndex{3}\): Flume examples of supercritical, critical, and subcritical flow as well as examples of hydraluic jumps and how obstructions interact with different types of flow.

Sediment Entrainment

Fluid dynamics matter because fluids (air, ice, and water) are important to sediment transport. Whether or not one of these flowing fluids gets things moving is determined by the balance between forces that are preventing movement (gravity, friction, cohesion) and the forces that are contributing to movement (drag, lift forces c/o Bernoulli’s Principle).

Video \(\PageIndex{4}\): An overview of Bernoulli's Principle with numerous examples.

If we were engineers, or just really interested, we could sit down and calculate these forces if we knew the properties of the liquid (shear stress, fluid viscosity) and the particle (shape, size, and density). But, we are not going to bother with that because many of them are difficult to predict or impractical to predict in nature.

Instead, we will rely on the Hjulstrom diagram to give us a sense of whether or not things are being eroded or deposited. This plot of grain size versus velocity is useful because it allows us to make some basic predictions based on particle size and flow velocity. Overall, what you can see is that it takes faster flow to move bigger particles … except for the case of mud where sediment cohesion makes particles stick together.

Figure \(\PageIndex{3}\): The Hjulstrom diagram can be used to predict the interaction between flowing water and sedimentary particles (Figure 13.16 by Steven Earle; CC BY 4.0)

Additional Readings and Resources

• Froude and Reynolds Number: https://www.geological-digressions.com/fluid-flow-froude-and-reynolds-numbers/
• Settling Velocity: https://www.geological-digressions.com/fluid-flow-stokes-law-and-particle-settling/
• Calculations using Stoke's Law: https://stormwaterbook.safl.umn.edu/sedimentation-practices
• Explanation of the Hjulstrom Diagram: https://www.geological-digressions.com/fluid-flow-shields-and-hjulstrom-diagrams/