Course:GEOBrefGuide/Geomorphology/Hydraulics

Introduction to Open Channel Hydraulics

In order to consider fluvial landforms from a process point-of-view, it is first necessary to consider the laws governing the flow of water. These laws are typically covered under the broad heading of open channel hydraulics, and they have been determined through theoretical and empirical means, mostly by hydraulic engineers. There are a number of key concepts, including:

• total stream power (Ω)
• unit stream power (ω)
• shear stress (τ) and shear velocity (u^*), which are basically the same thing
• mean velocity (u)
• Froude number (F_r)
• Reynolds number (Re)
• Reynolds particle number (Re^*)

We will define and explore all of these terms, as well as the equations used for estimating them. They are the basic building blocks used to understand fluvial geomorphology.

Continuity

Before getting into the key concepts, it is useful to define the relation between the geometric variables and stream discharge. The most basic relation is that which relates the mean velocity of the stream (u, in m/s) to the total discharge being carried by the stream (Q, in m ³/s) using the cross sectional area of the flow (A, in m²).

$Q=A\cdot u$

The cross sectional area can be further subdivided into a horizontal and a vertical component, one of which is measured, the other being estimated. If we divide A by the width of the water surface (W, in m), we get an estimate of the average water depth, referred to as the mean hydraulic depth (d, in m). Substituting these variables into the equation above gives:

$Q=W\cdot d\cdot u$

which is referred to as the continuity equation. This is the key equation that constrains many others, since it is really stating that the width, depth, and/or velocity with increase/decrease as discharge increases/decreases.

There is another possible way of subdividing the cross section area of flow: if we divide A by the total length of the channel bed and banks, starting at one water's edge and moving across the stream to the other (which is called the wetted perimeter, P, in m), we get a quantity that is similar to but slightly less than d. This quantity is called the hydraulic radius (R, in m). We can also use these variables to define a continuity equation:

$Q=P\cdot R\cdot u$

but this version is seldom if ever used by fluvial geomorphologists.

To summarize, the dimensions of the water-filled part of the channel change with discharge; the mean velocity is given by the ratio u=Q/A; and the cross sectional area can be related to either the channel width and depth (W, d) or to wetted perimeter and hydraulic radius (P,R).

Stream Power

The loss of potential energy per unit length of time (ΔE_p / Δt) determines the rate at which geomorphic work can be done. This is referred to as stream power. In some cases, geomorphologists use the total stream power that a river has (called the total stream power) and in others it is more useful to consider the stream power acting on a 1 m by 1 m area of the bed of the stream (called the unit stream power).

Total stream power

Recall that the loss of potential energy per unit length of stream channel is proportional to the change in elevation.

${\frac {\Delta E_{p}}{\Delta x}}=m\cdot g\cdot {\frac {\Delta z}{\Delta x}}$

where the mass, m, per unit length of channel depends on the width and average depth of the river (W, d), as well as the density of water (ρ). Thus, the equation becomes:

${\frac {\Delta E_{p}}{\Delta x}}=W\cdot d\cdot \gamma \cdot {\frac {\Delta z}{\Delta x}}$

where γ = ρg. To turn ΔE_p / Δ x into a loss per unit rate of time, we simply need to multiply both sides of the equation by Δx / Δt, as follows:

${\frac {\Delta E_{p}}{\Delta x}}\cdot {\frac {\Delta x}{\Delta t}}=W\cdot d\cdot \gamma \cdot {\frac {\Delta z}{\Delta x}}\cdot {\frac {\Delta x}{\Delta t}}$

On the left-hand side, the Δx terms cancel, leaving ΔE_p / Δt, which is referred to as the total stream power, and represented using the symbol Ω (measured in W/m). On the right-hand side, we realize that the term Δz / Δx is simply the slope or gradient of the stream channel, which is represented by the symbol S. The term Δx / Δt is the average velocity for the flowing water, represented by u. Making these simplifications and substitutions, we get:

$\Omega =W\cdot d\cdot u\cdot \gamma \cdot S$

Referring back to the continuity equation above, it is clear that the geometric terms (Wdu) simply represent the discharge carried by the stream. Thus, total stream power can be most easily predicted knowing nothing more than the gradient of the river (S) and the discharge the river happens to be carrying (Q):

$\Omega =\gamma \cdot Q\cdot S$

Unit stream power

The stream power acting on a 1m by 1 m area of the channel bed is simply given by dividing Ω by the width of the stream. This is called the unit stream power (ω, in W/m²).

$\omega ={\frac {\Omega }{W}}$

$\omega ={\frac {\gamma \cdot Q\cdot S}{W}}$

$\omega =\gamma \cdot q\cdot S$

where q is called the specific discharge and can be estimated at any point from the local depth (d) and velocity (u).

$q=d\cdot u$

So, if we know only the width of a channel and the discharge, we can estimate only an average value for ω that represents the average rate of potential energy loss per unit area of the channel bed. But, if we have measured the local depths and velocities during, for example, a stream gauging exercise, we can estimate ω at any point across the channel where d and u have both been measured.

Flow resistance

It is the friction between the flowing water and the bed and banks of the stream that controls how fast water can flow at a given stream gradient. The depth of the flow matters, too, since the further a molecule of water is from the bed or banks, the less friction it feels and the faster it can move. Since friction takes place everywhere along the wetted perimeter, P, it is most accurate to think of flow depths in terms of the hydraulic radius, R, rather than the mean hydraulic depth, d. The most common means for expressing the relation between velocity, depth, slope and flow resistance is the Manning equation:

$u={\frac {R^{2/3}\cdot S^{1/2}}{n}}$

Here, n is an empirically determined flow resistance parameter called Manning's n. The value of n increases as the mean particle diameter of sediment on the bed of the river increases (D₅₀), and it also varies with the channel morphology.

For natural channels that have steep, boulder-covered beds and that carry relatively small discharges, n can be as high as 0.06 to 0.07. In natural streams with lower gradients, large discharges and beds composed of sand or fine gravel, n will range from 0.025 to 0.035.

Fluid shear forces

Vertical velocity distribution and shear velocity

So far, we have addressed the means by with the average velocity is defined and estimated. However, at at single location across the stream, velocity varies systematically with distance from the bed. The variation of velocity with distance from the bed is called a velocity profile. The water flows most slowly near the bed, and increases with distance from the boundary until it reaches the mean free velocity. The region where the velocity changes continuously is called the boundary layer, and this is where shear stresses are generated. In many streams where the flows are relatively shallow, the boundary layer extends right to the surface of the water, and the flow never reaches the mean free velocity.

Within the boundary layer, the variation in velocity follows the log law of the wall:

$u(z)={\frac {u^{*}}{\kappa }}\ln {\frac {z}{z_{o}}}$

where κ is von Karman's constant (κ ≈ 0.4), z is distance measured upwards from the bed surface, u^* is the shear velocity, which is related to the diffusion of momentum from the flow towards the bed, and z_o is the height above the bed at which u = 0 m/s. The term z_o is related to the size of the sediment found on the bed and is called the roughness height.

The force that acts on the bed depends on how quickly momentum is being diffused towards the bed. In a flow, forces are transmitted been molecules that slide over each other but which exhibit some molecular attraction or stickiness. Thus the shear force of gravity acting to pull water down the slope is transmitted from molecule to molecule until it is ultimately transferred to the bed of the stream. The size of the force acting on the bed depends on the molecular viscosity of the fluid (i.e. the stickiness) and how fast the molecules are moving past each other (which is essentially the shear velocity, u_*). The shear force per unit area of the bed is given by:

$\tau =\rho \cdot u^{*\,2}$

Therefore, by measuring the velocity profile at a given point, we can use the data to estimate the shear velocity, u^*. Then, we can estimate the shear stress, τ, from the estimate of the shear velocity. Im this way, we could in theory estimate τ at a large number of places within the channel for a given flow, and then map the distribution of shear forces acting on the bed of the channel. This is seldom possible, given the difficulty of collecting enough data before flows change. While this kind of detailed information of the forces acting on the channel bed is highly desirable, fluvial geomorphologists often have to settle for estimates of the average shear stress acting on the channel boundary, which are more easily obtained.

Average boundary shear stress

If we think of a a length of stream channel down which water is flowing at a constant rate, we can immediately make the following observations using the derrivation of the infinite slope equation as a guide:

• the force of gravity acting on the is given by $F_{g}=m\cdot g\cdot \sin {\beta }=A\cdot L\cdot \rho \cdot g\cdot \sin {\beta }$
• this force must be opposed by an equal and opposite force if the mass of water is not accelerating (which we have said it is not)
• the opposing force is the shear stress generated at the interface between the flowing water and the bed, which is equivalent the the average shear stress times the surface area over which shear stress is being generated, so $F_{s}=\tau \cdot P\cdot L$

If we set $F_{g}=F_{s}$ and then rearrange to isolate τ, we get:

$F_{s}=F_{g}$

$\tau \cdot P\cdot L=A\cdot L\cdot \rho \cdot g\cdot \sin {\beta }$

$\tau ={\frac {A}{P}}\cdot \rho \cdot g\cdot \sin {\beta }$

Now if we recall that R = A / P and γ = ρg we can further simplify the equation. Also, for low gradients, we know that sin(β) ≈ S, and since river gradients are usually less than 0.02 m/m, we can write:

$\tau =\gamma \cdot R\cdot S$

remembering, of course, that this is only appropriate for estimating the average shear stress acting along the boundary, and that it does not tell us anything about the distribution of shear stresses across the channel.

Flow indices

Reynolds number

Most of the potential energy expended in a stream does not end up transporting sediment downstream. Most of it is turned into turbulence within the flow. Turbulent flows only occur if the mean flow velocity (u) is high enough, the flow is deep enough and the viscosity of the flow is low enough. The Reynolds number (Re) is a dimensionless ratio that is related to the degree of turbulence in a fluid flow. The equation is:

$Re={\frac {\rho \cdot v\cdot d}{\mu }}$

where μ is the fluid viscosity and d is the flow depth. When Re < 500, the flow is purely laminar and there is no turbulence. When Re > 2000 the flow is fully turbulent. In laminar flow, momentum diffuses through the flow as a result of just the molecular viscosity, while in turbulent flow, eddies carry high and low energy packets upwards and downwards in the flow, adding bulk flow momentum transfers to those associated with viscosity.

Froude number

In steep, fast flowing streams things called hydraulic jumps occur. At the jump, the flow transitions from a shallow flow with a very high velocity to a deeper flow with a low velocity. Both upstream and downstream of the jump the specific discharge, q, remains constant so that (ud)_upstream = (ud)_downstream. For the upstream section, the velocity is faster than the speed of a gravity wave moving through the flow ($u_{g}={\sqrt {gd}}$). A gravity wave is generated by, for example, throwing a stone into a pond, wherein the gravity wave takes the form of the ripples migrating away from the point-of-impact. In fluids, information about the presence of obstacles protruding into the flow (such as a bridge pier or a large boulder) is transmitted through the flow as a gravity wave, so for the upstream section, where the stream velocity is greater than the speed of a gravity wave, the flow does not see obstacles downstream until it crashes into it. This kind of flow is called supercritical flow. Downstream of the jump, the flow is deeper and slower, and the stream velocity is always less than the speed of a gravity wave. In this case, the flowing water feels the presence of an obstacle before it actually makes contact with it. This is called subcritical flow. The equation that is used to determine if the flow is supercritcal or subcrtical is:

$Fr={\frac {v}{\sqrt {gd}}}$

where Fr is the Froude number. For supercritical flow, Fr > 1, and for subcritical flow, Fr < 1 (Fr =1 is associated with critical flow, where the stream velocity and wave speed are exactly equal). In most streams, the flow is usually subcritical.

Interestingly, the occurrence of hydraulic jumps is one phenomenon that can be easily predicted from hydraulic theory. Consider the potential sources of energy for an element of water in a stream: it possesses potential energy (related to its elevation, z), energy due to the pressure force of the overlying water (related to the water depth, d) and due to the motion of the particle (related to u). The total energy for a 1 m by 1 m by 1 m cube of water, moving at velocity u is:

$E_{total}=\rho \cdot g\cdot z+\rho \cdot g\cdot d+{\frac {\rho \cdot v^{2}}{2}}$

If we divide the entire expression by the unit weight of water (ρg), then each component can be expressed as a length scale (called `heads', in m):

$H=z+d+{\frac {u^{2}}{g}}$

where H is called the total head, z is called the elevation head, d is the pressure head, and $u^{2}/2g$ is the velocity head.

In the vicinity of a hydraulic jump, we can assume that the change in elevation is relatively small, and so we can ignore the potential energy term. The remaining terms are referred to as the specific energy ($E=d+u^{2}/2g$, in m) of the flow. Over short distances, E is effectively constant. Now, remember that $q=Q/W=d\cdot u$. Since q is also constant (as long as the width of the stream does not change), we can express E as a function of one variable (either u or d) and a couple of constants (g and q). In the equation below, we have substituted u=q/d into the specific energy equation:

$E=d+{\frac {q^{2}}{2gd^{2}}}$

$E-d={\frac {q^{2}}{2gd^{2}}}$

$(E-d)\cdot d^{2}={\frac {q^{2}}{2g}}$

Since this equation is a cubic equation, there are three solutions for a given value of E, two of which are `real' solutions with positive d values, and the other is a mathematical artifact (d < 0). One of the real solutions corresponds to supercritical flow, having a small d value and a high u value (but the same q and same E). The other solution has a large d value and a smaller u (q and E again are constant). A hydraulic jump is what happens when one hops from one solution (i.e. the supercritical one) to the other. It does not happen when transitioning from the subcritical flow to the supercritical flow. The reason a jump occurs when transitioning from one solution to the other but not when transitioning in the opposite direction as to do with conservation of momentum. In fact, in order to conserve momentum and specific energy at the same time it is necessary that energy be dissipated at the hydraulic jump as turbulence, so in fact E is smaller downstream of a jump than upstream of it, even if momentum is conserved.

Reynolds particle number

Another important flow index expresses the flow conditions near the bed to the size of grains found on the bed. The Reynolds particle number (Re^*) is a dimensionless ratio with the same form as that for predicting the Reynolds number, but with different characteristic velocities and length scales. Referring to the equation for the Reynolds number and thinking about the typical velocity profile in the stream, it is clear that since both u and d get smaller as we approach the bed, Re also gets smaller, and since the flow becomes laminar for Re < 500, there should be a very thin layer of laminar-flowing fluid at the boundary: this is called the laminar sub-layer (δ, in m). The Reynolds particle number considers how fast the velocity is changing near the bed, as measured by the shear velocity, $u^*$, which replaces u in the equation, and it considers only the flow behavior around the particles making up the bed, so the particle diameter (D, in m) replaces the flow depth, d. The equation is:

$Re^{*}={\frac {\rho u^{*}D}{\mu }}$

When Re^* > 70, then u^* is large enough (and μ is small enough) that the laminar sub-layer is so thin that particles with diameter D protrude above it, into the turbulent part of the boundary layer, where they are exposed to shear forces due to both molecule-to-molecule transfers of momentum and bulk momentum transfers due to eddies that introduce a vertical component of the flow (i.e. D > δ). These conditions are referred to as hydraulically rough flows. When Re^* < 10, then particles of diameter D are entirely enveloped within the laminar sub-layer, where they are protected from turbulent eddies (i.e. D < δ). These conditions are referred to as hydraulically smooth flows. As we will see, Re^* plays and important role in determining the relative ease with which a particle can be entrained.