Basic Mathematical Concepts

The basic relations of open-channel flow and sediment transport are derived from the fundamental laws of classical physics, particularly the following:

Conservation of mass: Mass is neither created nor destroyed.

Newton’s laws of motion: 1) The momentum of a body remains constant unless a net force acts upon the body (conservation of momentum). 2) The rate of change of momentum of a body is proportional to the net force acting on the body, and is in the same direction as the net force. (Force equals mass times acceleration.) 3) For every net force acting on a body, there is a corresponding force of the same magnitude exerted by the body in the opposite direction.

Laws of thermodynamics: 1) Energy is neither created nor destroyed (conservation of energy). 2) No process is possible in which the sole result is the absorption of heat and its complete conversion into work.

Fick’s law of diffusion: A diffusing substance moves from where its concentration is larger to where its concentration is smaller at a rate that is proportional to the spatial gradient of concentration.

Equations based on these relations are developed by first stating the appro-priate fundamental law(s) in mathematical form, incorporating the boundary and (if required) initial conditions appropriate to the situation, and then applying the principles of algebra and calculus. These mathematical formulations require two assumptions that are not physically realistic, but that fortunately lead to physically sound results: 1) the fluid continuum, and 2) the fluid element. Formal mathematical developments also require the specification of a formal system of spatial coordinates (usually the three mutually perpendicular Cartesian coordinates), and may also involve time as an additional dimension. These concepts are presented here.

4.1.1 Fluid Continuum

The techniques of calculus—taking derivatives and integrals—are essential tools for expressing basic physical principles in mathematical form. Underlying the application of these techniques to problems of fluid flow is the concept of the fluid continuum:

To apply the mathematical concept of “taking limits,” which underlies the definitions of derivatives and integrals, we must imagine that the bulk properties (density, pressure, viscosity, velocity, etc.) exist even as we consider infinitesimally small regions of the fluid. In reality, of course, fluids are made of discrete molecules, and the bulk properties are not defined at the molecular scale. Fortunately, the fiction of the fluid continuum serves us well for the purposes of earth sciences and engineering.

4.1.2 Fluid Element

Fluids are also continua in the sense that, in contrast to solids, there are no physical boundaries separating the elements of a flow. Thus, another useful fiction commonly invoked in analyzing fluid-flow situations is that of the fluid element

BASIC CONCEPTS AND EQUATIONS 139 or fluid particle: “Any fluid may be imagined to consist of innumerable small but finite particles, each having a volume so slight as to be negligible when compared with the total volume of the fluid, yet sufficiently large to be considered homogeneous in constitution” (Rouse 1938, p. 35). Each particle at any instant of time has its own particular velocity and other properties, which generally vary as it travels from point to point.

4.1.3 Coordinate Systems

Precise mathematical descriptions of objects in space require specification of a coordinate system. The two coordinate systems used in this text are illustrated in figure 4.1. We use the standard orthogonal Cartesian x-, y-, z-coordinate system when focusing on fluid elements and other phenomena at the “microscopic” scale (figure 4.1a). We will often restrict our interest to two dimensions, with the z-axis oriented vertically and the x-axis directed in the “downstream” direction.

When examining flows in channels at the more macroscopic scale, we will usually use a two-dimensional coordinate system, replacing the (x, y, z) coordinate directions with (X, y, z). We maintain the z-axis vertical and the X-axis downstream, but because the channel bottom will generally be sloping at an angle 0 (measured positive downward from the horizontal), the X-axis will make an angle of /4+ 0

(90^◦+ 0) with the z-axis (figure 4.1b). The y-axis is oriented normal to the X-axis with y= 0 at the channel bottom, so distances in the y-direction are distances above the bottom. Distances measured along the y-axis are related to those measured along the z-axis as

y= (z − z0)· cos 0, (4.1)

where z0is the elevation of the channel bottom above an arbitrary elevation datum.

In a few instances, we define a “depth” (i.e., distance below the surface) direction as h≡ Y − y, where Y is the height of the surface above the bottom.

For two-dimensional mathematical representations of channel cross sections (figure 4.1c), we use w for the cross-channel direction, generally taking w= 0 at the channel center. The vertical direction is represented by z.

In this text, we will assume that coordinate systems are fixed relative to points on the earth’s surface, and that those points are stationary. In reality, points on the earth are moving through space and, more significantly, rotating due to the earth’s rotation around its axis. This rotation gives rise to the Coriolis effect, which introduces accelerations to objects moving with respect to a fixed coordinate system. These accelerations increase from zero at the equator to a maximum at the poles. However, as we will show in chapter 7, the Coriolis effect becomes significant only for very large-scale flows such as ocean currents, and it is safe to ignore the effect at the scale of river flows.

Accelerations are also induced due to momentum when fluid elements follow curved paths in a fixed coordinate system. These accelerations are usually treated as centrifugal force and can be important in river flows, as discussed in chapters 6 and 7.

(a)

z

y

x = 0, y = 0, z = 0 x

(b)

z y

θ₀ z = z₀

X y = 0

z = 0 Elevation datum

Figure 4.1 Coordinate systems used in this book. (a) The standard Cartesian coordinate system with x-, y-, z-axes orthogonal. The z-axis is usually oriented vertically, and the x-axis is usually directed in the principal flow direction (downstream). (b) The coordinate system used for two-dimensional flow macroscopic flow descriptions. The z-axis is oriented vertically with its 0-point the elevation of an arbitrary datum. The X-axis is directed in the principal flow direction (downstream). The y-axis represents distance above the bottom. It is oriented normal to the X-axis and makes an angle ₀with the z-axis; y= 0 at the channel bottom. (c) For channel cross sections, w represents the horizontal cross-channel direction, with w= 0 usually at the channel center. The z-axis is oriented vertically with its 0-point usually at the elevation of the deepest point of the channel.

BASIC CONCEPTS AND EQUATIONS 141

(c)

z

w

w 0

Figure 4.1 Continued

4.1.4 The Lagrangian and Eulerian Viewpoints

Problems of fluid flow can be analyzed in two formal viewpoints: In the Lagrangian¹ viewpoint, we follow the path of a fluid particle as it moves through space. In the Lagrangian approach the location of an individual fluid element is a function of time.

Thus, for an element that is at location x0, y0, z0at time t0, its subsequent locations are functions of its original location and time, t:

x= f1(x0,y0,z0,t), y= f2(x0,y0,z0,t), z= f3(x0,y0,z0,t). (4.2) In the Eulerian viewpoint, we observe the behavior of fluid elements as they pass fixed points. Thus, in the Eulerian approach the fluid properties are functions of fixed location coordinates and time:

qx= f1(x, y, z, t), qy= f2(x, y, z, t), qz= f3(x, y, z, t), (4.3) where qx, qy, qz represent fluid properties (e.g., velocity, acceleration, density) that may vary in the three coordinate directions.

Comparing equations 4.2 and 4.3, we see that in the Eulerian approach the spatial coordinates, along with time, are independent rather than dependent variables. This is usually the simpler way of analyzing a flow problem and is the one we will most often use herein. However, it is sometimes possible to convert time-varying flows to simpler time-invariant flows by switching from a Eulerian to a Lagrangian viewpoint (e.g., in considering the settling of sediment particles, or the passage of a wave along a channel).