# 3. Fluid Flow 2

## Key Concepts from Lecture 2

### Reynolds Number

Reynolds number predicts the extent of turbulence in a fluid based on how fast the fluid is flowing, the geometry of the flow (how deep and wide it is, etc.), and the density and viscosity the of the fluid.

$Re = \dfrac{\text{fluid inertial forces}}{\text{fluid viscous forces}} = \dfrac{l \times u \times \rho}{µ} \tag{3.1}$

where the variables are

• flow velocity ($$u$$),
• characteristic length ($$l$$) which represents flow geometry, say river depth,
• fluid density  ($$\rho$$), and
• fluid viscosity ($$µ$$).

For the version of the equation we are using, turbulent flow has Re is greater than 2000 and laminar flow has Re is less than 500. Flow with Re between 500 and 2000 is transitional and has some characteristics of laminar flow, but some turbulence as well.  (Note that different versions of the Reynolds equation are used for different flow geometries and different equations have different numerical boundaries between laminar, transitional, and turbulent flows.)

### Boundary Layer and Laminar Sublayer

There is boundary layer at the edge of every flow where flow speed decreases due to friction. Within the boundary layer, right next to the surface, the flow speed is very low, creating a laminar sublayer.

## Sediment Transport

### Bed Shear Stress

The boundary layer determines the amount of “Bed Shear Stress” which corresponds to the forces that tend to roll particles along the bed and the pressure differences above and below the grain which tend to lift them off the bed. Bed shear stress is related to the thickness of the laminar sublayer.  The narrower it is, the more bed shear stress. It also depends on the slope.  If the slope is steep, gravity helps pull grains down the slope, increasing bed shear stress. Also, the roughness of the bed is a factor.  A rough bed deflects flows and increases turbulence, which increases the bed shear stress, particularly in places where flow is directed into the sediment and the boundary layer is compressed. (See http://en.wikipedia.org/wiki/Sediment_transport#Bed_shear_stress for more detailed information.)

### The Bernoulli Effect

A pressure difference “pulls” grains off the bed.  The pressure difference comes from a difference in water (or air) speed above and below the grain.  As water flows faster, there are fewer collisions between the water and a surface it flows over than there are between standing water and a similar surface.  Pressure is due to collisions.  Thus, fewer collisions means lower pressure.  The upstream side of a grain experiences the most collisions because the water is flowing into it.  The downstream side experiences the fewest collisions, and the sides of the grain experience fewer collisions where flow is faster and more where the flow is slower.  The net result is a low pressure zone above and slightly downstream of a grain.  If the force exerted by this pressure difference is larger than the force of gravity, the grain will lift off the bed.  This lift due to the pressure difference is the Bernouli Effect.

#### Which Grains Move?

Which grains get entrained in the flow depends on their size and density (how much they weigh) because that determines the force of gravity holding them down.  It also depends on the shape of the grain.  One with a large area to experience the low pressure (like a plate) will be more susceptible to being picked up than a round grain of the same mass (although flat grains may see a smaller flow difference from top to bottom if the boundary layer is thick, and thus flat grains may experience a lower Bernoulli Effect per unit area.)  The other thing that really matters is the position of a grain relative to surrounding grains.  If a grain is sandwiched between larger grains, i.e. in their flow shadows, it will not experience as big a pressure difference as if it is on a flat surface.  Also, if a grain is upstream of a big grain, it has to be lifted over it, so it has to experience enough force to lift it high into the flow.  Thus, things can complicated if you are trying to predict the behavior of a specific grain.  However, experiments and theory provide statistically meaningful predictions for how grains behave on average.

Two things can happen once a grain is lifted into the flow:  1) it can fall back down or 2) it can stay there.  It depends on how quickly the grain settles out versus how turbulent the water is (back to Re...).  Bedload refers to the grains that are transported along the sedimentary bed, e.g. grains that are rolling and being lifted off the bed, but they fall back quickly.  The name bedload comes from the fact that the grains moving by traction and saltation never get too far from the bed and “load” is an engineering term for the amount of sediment transported by a river.  Rolling grains are in traction.  Grains that are pulled off the bed with the Bernoulli effect but are large enough that gravity causes them to fall “quickly” back to the bed are said to be saltating.  (The word saltating refers to the way salt from a salt shaker bounces when it is shaken onto a hard surface.  The word is derived from a Latin word meaning dance.)  Bedload grains are the ones that form sedimentary structures in flowing water.

Here is a playlist with movies related to sediment transport: Sumnerd’s Sed Transport Playlist

Suspended sediment consists of grains that are light enough that they do not settle out of the water; the turbulent bursts of water keep them in the flow (see brown water in the photo).  The more turbulence in the water, e.g. the higher the Reynolds number, the larger the grains in suspension will be.  The upward motions of turbulent flow are faster than the rate these grains settle, so gravity is counteracted and they stay “floating” in the water even though they are denser than the water. Very small grains do not settle out of flows unless the Reynolds number is low, which means that the flows need to be standing or very shallow.

YouTube video of white clay in a turbulent flow in a flume: http://tinyurl.com/78kg3z  The pulsing in the flow is (probably) due to the pump that is making the water flow.

### Hjulstrom Diagram

The flows that are required to pick up grains of certain sizes have been extensively studied in experiments and the results are plotted in Hjulstrom diagrams.  Hjulstrom diagrams show grain entrainment on a plot of log grain size versus log flow speed.  This diagram shows the areas where grains of different sizes are left on the bed, where they get moved sometimes (this is the gray zone), and where they get lifted up often and eroded away.  Note that larger grains require higher flows - in general.  The water speed that is required to transport a grain is call the critical velocity.  This is important.  If there is gravel in a sedimentary deposit, you can say that the water flow had to be above the critical threshold for it to get there!  That might require a fast flowing river or strong wave action, thus, a large part of narrowing down the depositional environment has already been done!

Here is the Hjulstrom Diagram we will use: (https://commons.wikimedia.org/wiki/File:Hjulstr%C3%B6ms_diagram_sv.PNG)

The vertical axis is flow speed in cm/s (in Finnish!) and the horizontal axis is grain size in mm (in Finnish!); note that both axes are on a log scale. This diagram is for a flow depth of 1 m. If the flow speed and grain size are in the field labeled "Deposition", grains of this size will not be lifted into the flow, and if they are already moving, they will be deposited onto the sediment surface.  If the flow speed and grain size are in the field labeled "Transport", grains in motion are likely to continue in motion.  A few grains will be deposited and a few grains will start moving, but there will not be a significant change in the number of grains that move.  If the flow speed and grain size are in the field labeled "Erosion", more grains will be transported than deposited until the flow is transporting as many grains as it can, e.g. it is at its "carrying capacity" for sediment.

The boundaries for deposition, transport and erosion shift with changing flow depth.  For example, deeper flows can move larger grains at the same flow velocity because they are more turbulent:  $Re = \dfrac{l \times u \times \rho}{µ} \tag{3.1}$ and l is larger.  This is because deeper flows can have larger variations in flow speed and the laminar flow layers are very thin.  They can have bursts of very rapid flow relative to the average flow speed and these bursts can pick up larger grains.

Actual flow characteristics are much more complex in detail than just Hjulstrom diagrams, which summarize a lot of characteristics into two axes.  However, like a lot of people, we will use the diagram anyway, because it is very useful as a rule of thumb.  Just remember that it is not a completely accurate representation of what will happen - it represents a reasonable guess.

#### Silt and Clay Transport

Notice that for the small end of grain size, the speed of flow required for erosion actually increases. One reason small grains are hard to erode is that they tend not to stick up through the laminar sublayer; they are just too small.  Thus, thinner boundary layers are necessary to roll them or for the pressure differences to pick them up off the bed.  Also, the surfaces of clay minerals tend to be charged and the grains stick together.  This is most obvious when big clumps of mud stick to your shoes.  That just does not happen with sand (unless there is something gross in it).  The stickiness of the clay grains makes them difficult to erode, so faster water flows (a greater pressure difference or larger turbulent burst down to the sediment surface) are required to move them.  The smaller the grains, the more surface charges stick the grains together, thus the stronger the flow needed to erode them.  The stickiness of the clay grains also depends on the amount of water between them and the mineralogy, so there is a big gray zone where a clay may or may not erode.

In the Hjulstrom diagram, there is an interesting area where the flow is not strong enough to move any of the particles on the bed, but those that are in the suspended load do not settle out either.  This zone includes many of the waters on the surface of the earth. In flows with any velocity or that are very deep, Re is high enough to keep some clay in suspension.  Clay deposition usually occurs very slowly, e.g. when the rate of settling is just slightly faster than the average rate at which turbulence moves clay particles upward or when the clays clump together to form larger grains (which is common when fresh and salty waters mix).

## Miscellaneous Notes

A few more words about saltation:  Saltation is a very interesting and important process in sediment transport, because the force of the impact when the grains land tends to knock new grains up into the flow even if the flow is not fast enough to lift them with the Bernoulli Effect.  These new grains can kick up more grains when they land, etc.  This increases the rate of sediment transport above the amount the flow can lift grains off of the bed.  This is one of the causes of the gray zone in the Hjulstrom diagram at larger grain sizes.  Once saltation starts, it can trigger sediment transport that would not otherwise occur.

Deposition:  Deposition is the accumulation of grains.  If a flow starts slowly and gains speed, it will start to move larger and larger grains.  As it slows down, it can only move the smaller ones.  Deposition happens when a flow slows down and starts to leave grains on the bed.  The combination of changing average flow speeds and local variations in flow speed caused by topography on the bed give rise to very informative sedimentary structures – including cross stratification - which are extremely useful for interpreting depositional environments.