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Basic Principles and Flow Classifications

Hydraulic Engineering has served humanity all through the ages by providing drinking water and protective measures against floods and storms. In the course of history, it has made the water resource available for human uses of many kinds. Biswas (1970) chronicles contributions since Hammurabi (c. 1700 BCE) hydraulic engineering over the centuries. In a survey of the University of Georgia Libraries' holdings under "land-use change," some 8000 articles discuss facets of the hydrologic cycle and associated runoff. Simons and Senturk (1992) provide a synopsis of contributions to sediment transport science in streams that date back to work in China to date back to 4000 BCE. Students who seriously pursue open channel hydraulics and sediment transport should explore works such as those mentioned. Management of the world's water is a complex task, and both its scope and importance continue to grow as we strive for sustainable stewardship of our abode.

Over time humanity has not only diverted and used the waters of the world for its purposes but, by engaging nature into its service, has turned deserts into fertile land. Natural habitat is threatened in more and more parts of the world by an ever-growing human population. Thus, long-term needs are food, water, shelter, and an aesthetically pleasing, healthy, nurturing environment.

Open channel flow is, in brief, a flow where the fluid has a free surface, where the free surface of the flow is subject to atmospheric pressure. Problems covered include flow in a conduit when the conduit is not full, such as in a storm sewer. The primary fluid of interest is water, although any fluid could, in principle, be addressed. This chapter introduces concepts that are developed as we progress through the text.

Why do we consider open channels when one can simply take an earthmover and create a conveyance? The hydraulic engineer meets engineering, economic, and social objectives in the client's and society's best interests. Sound engineering should result in sustainable design. Figure 1.1 shows a channel under construction with a channel design that conveys water while meeting aesthetic and sustainability goals.

Here are some definitions:

• Ditch - an excavated water conveyance often installed without extensive advanced engineering design.
• Natural channel - watercourses that exist naturally, such as gullies, brooks, rivers, streams, or estuaries. These are often analyzed to determine flows associated with high-water marks.

  Figure 1.1 A trapezoidal waterway under construction

  (Source: Photo courtesy of Mr. Greg Jennings).

• Ephemeral channel - a natural channel that does not continuously flow.
• Perennial channel - the natural channel that continually flows.
• Artificial channel - watercourse developed by humankind such as navigation, irrigation, drainage, or closed conduit (e.g., flow does not fill the entire channel such as a sewer or culvert) channels. Hydraulically designed conveyances with attributes strategically chosen to meet the client's engineering, economic, and social objectives (Graf and Altinakar 1998). Designed channels not continuously flowing are intermittent.

We address the hydraulic engineering needed to satisfy social, economic, and engineering needs in this text. We focus mainly on one-dimensional solutions in this introductory text. Once solved using tabular solutions, the one-dimensional problems (along the channel) are easily solved with spreadsheets and other equation processing software. A commonly used public domain software, HEC-RAS, can, in some cases, facilitate two-dimensional solutions.

Fluid Mechanics Foundations

Fluid statics is a crucial limiting case. It describes fluid pressures and forces when the fluid velocity is zero. The pressure at a point is the product of fluid unit weight, ?, and depth, symbolized as d, b, or y. The unit weight in SI and imperial units is 9810?N/m3 or 62.4?lb/ft3. Figure 1.2 shows a hydrostatic distribution on a channel wall and bottom. Figure 1.3 shows the hydrostatic forces on a sloping wall where one partitions the effects between vertical and horizontal vector sums. Acceleration causes deviations from the purely hydrostatic pressure distribution. Sluice gates and transitions due to structures and slope changes result in acceleration. Thus, the pressure is not hydrostatic near sluice gates and other facilities.

One can easily show that the effective center of force is 1/3 up from the channel bottom. The 1/3 rule applies when the wall or gate object is rectangular. Symmetric gates of other shapes have force centers at the center of pressure, which deviates from the 1/3 point. In general, the center of pressure is computed using the following equation:

Figure 1.2 Free body diagram of a static fluid.

Figure 1.3 Fluid forces exerted on a submerged gate.

where IG is the moment of inertia (b * y3/12 for a rectangular gate where b is length and y = depth), is the depth to the centroid of the object (half the distance to the bottom for a rectangle), A is the area of the port or gat object (length b ×?depth y for a rectangular gate), and yp is the distance below the surface to the centroid where the total hydrostatic force is concentrated.

It is easily shown that yp is 2/3 the depth from the top or 1/3 of the bottom for the rectangular gate. Equation 1.1 gives the distance along the slope for non-vertical gates. One can separate the forces into horizontal and vertical vector components and apply the 1/3 rule to the vertical component, giving a similar value to Equation 1.1. The value of Equation 1.1 comes to the fore when the gate is nonrectangular. One may then consult a statics text for the moment of inertia for the shape in question. Nonsymmetric gates (along the length b-axis) have a product of inertia, which shifts the center of pressure a distance from the centroid along the width b of the gate. We are not concerned with nonsymmetric gates. One may apply fundamental statics analysis to compute forces to secure closed gates in a channel with ponded, static conditions. A spreadsheet is provided which analyzes the forces required to fasten a symmetric gate on sloping walls.

A partially opened gate obviously does not represent a static condition. We evaluate forces associated with moving water in our chapter on rapidly varied flows. As in closed conduits flowing full, flows in open channels may be laminar or turbulent. The flow is laminar when viscous forces dominate inertial forces in determining flow behavior. Flows are turbulent when inertial flows dominate viscous forces. The Reynolds number, is expressed as follows:

where R is the Reynolds number (-), ? is the kinematic viscosity ((L/T2), typically 1.93E-06?m2/s or 1.93E-05?ft2/s), µ is the dynamic viscosity (FT/L2, typically 3.75E-05?lbf s/ft2 or 1.79E-03?N s/m2), ? is the fluid density (M/L3), typically 1.94 slugs/ft3 or 1000?kg/m3, L is the characteristic length, typically depth (L), and V is the velocity (L/T).

Units for viscosity are quite varied, depending on the usage of force or mass units. Dynamic viscosity may be expressed in mass units as M/(LT). One may perform an internet search to find these expressions in desired units. We include a spreadsheet showing viscosity and density as a function of temperature for SI and Imperial units. We generally ignore temperature effects in most open channel applications.

As in closed conduit flow, flows with Re?<?2000 are generally laminar, and flows with Re?>?10?000 are generally turbulent. The region 2000?>?Re?>?10?000 is a transition zone. Consider the Darcy-Weisbach formula given as follows:

where hf is the head loss (L), f is the friction factor (-), do is the diameter of the pipe (L), g is gravity (L/T2), and L is the length over which the head loss occurs (L).

Defining the slope as hf/L, do as 4R (where hydraulic radius R is more fully defined later), one may write Equation 1.3 as follows:

Simons and Senturk (1992) compare friction factor vs. Re pipes and channels with varying roughness heights. Their figure is reproduced in Figure 1.4. The Darcy-Weisbach friction factor may be related directly to the Manning state equation friction term in the turbulent zone. The concept of channel roughness is more fully developed in the next chapter.

Figure 1.4 shows notable similarities with the Darcy-Weisbach-Moody diagram for closed conduit pipe flow. Most flows generally occur in the regime where Re?>?4000, which enables one to relate the friction factor to roughness height. Most flows of interest involve turbulent flows, except for shallow sheet flows and flow found in natural...