# 4.4 Fluid Mechanics of Pipes and Open Channels

When a pipe is full, flow within the pipe can be assumed to be one dimensional along the axis of the pipe. Flow velocity at the pipe wall is zero and increases towards the center of the pipe. Experiments of the late 1850s on flow of water in straight cylindrical pipes indicated that the head loss along the pipe varied “*directly with velocity head and pipe length, and inversely with pipe diameter*” (Vennard and Street, 1975). Owing to the parabolic shape of the velocity distribution across the center of a circular pipe during laminar flow (Figure 34a), flow through the pipe can be approximated from the pressure gradient (that is, the difference in pressure on each end of a pipe, divided by its length) by using the Hagen-Poiseuille equation (Vennard and Street, 1975).

Equation 3 describes the relationship between pressure gradient and specific velocity based on the Hagen-Poiseuille equation for laminar flow in a full pipe.

(3) |

where:

∆p |
= | pressure difference between two ends of the pipe (ML^{−1}T^{−2}) |

L |
= | pipe length (L) |

r |
= | pipe radius (L) |

Exercise 11 invites the reader to visit the Wikipedia page discussing the Hagen-Poiseulle equation and the Poiseuille law and then consider the relationships and associated assumptions.

For turbulent flow in full pipes, an empirical equation developed by Henry Darcy and Julius Weisbach (the Darcy-Weisbach Equation) indicates that the pressure gradient is proportional to the square of the mean velocity (*V=**Q/A*) and a dimensionless friction factor as shown in Equation 4.

(4) |

where:

f_{D} |
= | Darcy friction factor (dimensionless) |

D |
= | hydraulic diameter of the pipe (L) (for circular pipe it is the pipe diameter, but for a non-circular pipe; where A is the cross-sectional area in L^{2}) |

The friction factor in Equation 4 is a function of the Reynolds number and the relative roughness of the pipe. The relative roughness of a pipe is usually defined by the ratio of roughness of the pipe wall and the mean height of the pipe. Under laminar flow conditions for a smooth circular pipe, *f*_{D} = 64/*Re*.

Exercise 12 invites the reader to substitute 64/*Re* and the equation for the Reynolds number into Equation 4 to confirm that this results in the Hagen-Poiseuille equation for a circular pipe.

The density and viscosity of water change with temperature. If density and viscosity are constant, many fluid flow problems can be solved with water level gradients rather than pressure gradients as shown in Equation 5.

(5) |

where:

g |
= | local acceleration due to gravity or gravity constant (LT^{−2}) |

The original Reynolds experiments were conducted with smooth glass pipes (Reynolds, 1883). By running multiple experiments with different diameter pipes and at different temperatures, he discovered that there was an upper and lower critical Reynolds number for all straight pipes and all fluids. Due to conservation of momentum, flow in a laminar state tends to stay laminar and flow in a turbulent state tends to stay turbulent. Thus, when the velocity of laminar flow gradually increases the flow becomes turbulent at a higher *Re* than the value at which flow becomes laminar when the velocity of turbulent flow gradually decreases. That is, a lower *Re* is required before the flow goes back to a laminar state. These are called the upper and lower critical Reynolds numbers (*U*_{R} and *L*_{R}, respectively).

It has been observed that between the *U*_{R} and *L*_{R}, the discharge in pipes is a function of mean velocity to a power greater than 1 but less than 2 (Vennard and Street, 1975). The *U*_{R} for smooth glass pipes in the original experiment was between 12,000 and 14,000 but has little practical use as most water pipes are manufactured from rougher material and/or are not straight. For more common pipes, the *U*_{R} value is smaller and depends on the roughness and shape of the pipe. A rougher pipe surface results in the onset of turbulence at smaller velocities and thus a smaller *U*_{R} and a curve or bend in a pipe also results in a lower *U*_{R}. According to Vennard and Street (1975) for practical purposes, the *U*_{R} likely falls between 2700 and 4000 for common pipe materials. The *L*_{R} denotes flow is in a laminar state. For circular pipes, if *Re* is less than 2100, flow is likely laminar and if *Re* is greater than 4000 flow is likely turbulent. Differently shaped conveyances result in different critical *Re*. For example, for flow between parallel plates, using plate spacing instead of pipe diameter, the *Re* below which flow is laminar is 1000; for wide, open channels using flow depth instead of pipe diameter, the *Re* for which flow is laminar is 500; and for flow around a sphere using sphere diameter instead of pipe diameter, the *Re* for laminar flow is generally close to 1. As with groundwater, the actual critical Reynolds number for pipes and natural channels is determined through experiments.

Exercise 13 invites the reader to consider how the irregularities of dissolution features impacts the onset of turbulent flow.

Most naturally occurring conduits in karst aquifers are not circular in cross-sectional shape (Figure 16, Figure 17, Figure 25, and Figure 26). In fluid mechanics some approximations are used to allow estimates based on equations for circular pipe geometry. Two important terms in open channel flow hydraulics are wetted perimeter and hydraulic radius. The terms are defined in the same manner whether they are used for a fully submerged pipe, an open stream channel or a karst aquifer conduit. The wetted perimeter is the length of conveyance wall on a cross-section perpendicular to flow that is fully wet from flow (Figure 37). The hydraulic radius is the ratio of the cross-sectional area to the wetted perimeter (for example, for each item of Figure 37 the area of the blue shape divided by the length of the red line). For karst conduits, estimates of *Re* are conducted using the equations for circular pipe that would have the same hydraulic radius as the conduit and this radius is used to calculate the diameter. The wetted perimeter and the effective hydraulic radius of karst conduit passages vary substantially as shown in Figure 38.

Exercise 14 invites the reader to calculate the hydraulic radius of the circular pipe and the concrete channel of Figure 37.

Open-channel flow is dependent on gravity and the slope, shape, and roughness of the conveyance. Flow in open channels can be laminar or turbulent, steady or unsteady. Steady flow is defined as when the velocity, pressure, and kinematic viscosity (density and temperature) of the flowing fluid remain constant through time at a cross-section. Unsteady or non-steady flow indicates that the fluid properties at a point change with time. Steady flow is like laminar flow, but not the same in that in laminar flow each particle moves along the same line at a constant velocity and no streamlines cross each other (Figure 34a). Thus, laminar open-channel flow would always be considered steady flow. Often stream flow can be turbulent, but if the average velocity remains constant with no change in pressure, density and viscosity of the water then this is steady-turbulent flow. It is beyond the scope of this section and even elementary fluid mechanics to cover the topic of unsteady flow.

Flow in a full pipe is different from open-channel streamflow or flow in partially full pipes. The primary difference is that the water surface in a stream or partially full pipe is exposed to atmospheric pressure over the entire water surface. Thus, flow is not related to pressure at the ends of the pipe and the friction factor of the pipe wall, because the pressure is the same across the entire surface. Flow in a full pipe can be laminar or turbulent. Figure 39 links to a video showing laminar flow in large karst conduits.

Exercise 15 invites the reader to consider, how wetted perimeter and the effective hydraulic radius for the two conduit passages of Figure 37 would be calculated under varying flow conditions, and how flow velocities would vary as the conduits fill.

Exercise 16 invites the reader to consider why we define three types of karst porosity, particularly with respect to the occurence of laminar and turbulent flow.