Basic Concepts and Equations
4.2 Kinematics and Dynamics
Relations that involve only velocities and/or accelerations (i.e., quantities involving only the dimensions length [L] and time [T]) are kinematic relations; those that involve quantities with the dimension of force [F] or mass [M] are dynamic relations.
Newton’s second law of motion, “force (F) equals mass (M) times acceleration (a),”
provides the basic link between kinematics and dynamics:
F= M · a, (4.4a)
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which also expresses the relation between the basic physical dimensions of force, [F], and mass, [M]:
[F] = [M] · [L T−2], (4.4b)
(see appendix A for a review of dimensions of physical quantities).
4.2.1 Kinematics 4.2.1.1 Velocity
The velocity in an arbitrary s-direction, us, is the time rate of change of the location of a fluid element:
us≡ds
dt, (4.5)
where ds is the distance moved in the time increment dt. Thus, velocity is a vector quantity with dimension [L T−1] that has direction as well as magnitude.
In the Eulerian viewpoint the direction can be specified by resolving the actual velocity into its components in the orthogonal coordinate directions (illustrated for two dimensions in figure 4.2) such that
ds dt = 1
cos x·dx dt = 1
cos y·dy dt = 1
cos z ·dz
dt, (4.6)
where x, y, z are the angles between the s-direction and the x-, y-, and z-directions, respectively. Defining the components of velocity in the three coordinate directions as
ux≡dx
dt, uy≡dy
dt, uz≡dz
dt, (4.7)
the magnitude of the velocity is
us= (ux2+ uy2+ uz2
)1/2. (4.8)
dz ds
•
dx
Figure 4.2 The distance ds traveled by a fluid element in an arbitrary direction in time dt can be resolved into distances parallel to the orthogonal x- and z-axes, dx and dz.
BASIC CONCEPTS AND EQUATIONS 143 Recall from section 3.3.4 that most open-channel flows are turbulent, and the velocities of fluid elements change from instant to instant and have chaotic paths (see figures 3.20 and 3.21). Thus, to be useful in describing the overall flow, the velocities discussed in this chapter—and in most of this text—are time-averaged to eliminate the fluctuations due to turbulent eddies; that is, they are the ¯ui quantities defined in figure 3.25.
Velocity is, of course, a central concern in fluid physics, and although it is a vector quantity, “knowledge of vector analysis is not essential to the study of fluid motion, for the variation of a vector may be fully described by the changes in magnitude of its three components” (Rouse 1938, p. 35). These changes—accelerations—are discussed in the following section.
4.2.1.2 Acceleration
Acceleration is the time rate of change of velocity, with dimension [L T−2].
Acceleration is also a vector quantity, and in the Eulerian viewpoint we write the accelerations for each directional velocity component separately. A change in the component of velocity in the i-direction, dui, where i= x, y, z is the sum of its rate of change in time at a point ∂ui/∂t times a small time increment dt, plus its rates of change in each of the three coordinate directions times short spatial increments in each direction, dx, dy, dz:
dui=∂ui
∂t · dt +∂ui
∂x · dx +∂ui
∂y · dy +∂ui
∂z · dz (4.9)
Acceleration in the i-direction is dui/dt, so from equation 4.9, dui
and using the definitions of equation 4.7, we can write the expression for acceleration in the i-direction as
Equation 4.11 gives the rates of change of velocity components ux, uy, uz for a fluid element at a particular spatial location and instant of time. These accelerations are the sum of the local acceleration and the convective acceleration:
Local acceleration is the time rate of change of velocity at a point, ∂ui/∂t.
If the local acceleration in a flow is zero, the flow is steady; otherwise it is unsteady.
Convective acceleration is the rate of change of velocity at a particular instant due to its motion in space, (∂ui/∂x)· ux+ (∂ui/∂y)· uy+ (∂ui/∂z)· uz. If the convective acceleration in a flow is zero, the flow is uniform; otherwise it is nonuniform.
Flows may be steady and uniform (no acceleration), steady and nonuniform (convective acceleration only), or unsteady and nonuniform (both local and
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convective acceleration); unsteady uniform flows (those with local acceleration only) are virtually impossible. Again, these definitions refer to the time-averaged velocities neglecting the fluctuations due to turbulent eddies.
4.2.1.3 Streamlines and Pathlines
A streamline is an imaginary line drawn in a flow that is everywhere tangent to the local time-averaged velocity vector (figure 4.3). If a flow is either steady or uniform, the streamlines are also pathlines; that is, they represent the time-averaged paths of fluid elements, neglecting motion due to turbulent eddies. In uniform flow, the streamlines are parallel to each other (figure 4.3c). Many of the basic relationships of open-channel flows are developed first for “microscopic” fluid elements and streamlines, and then integrated to apply to macroscopic flows.
4.2.2 Dynamics
4.2.2.1 Forces in Fluid Flow
The forces involved in open-channel flows are as follows:
Body forces: gravitational (directed downstream); Coriolis (apparent force perpendicular to flow); centrifugal (apparent force perpendicular to flow) Surface forces: pressure (directed downstream or upstream); shear (directed upstream)
Body forces act on all matter in each fluid element; surface forces can be thought of as acting only on the surfaces of elements, and are often expressed as stress— that is, force-per-unit area.
Gravitational and shear forces are important in all open-channel flows: Flow in open channels is induced by gravitational force due to the slope of the water surface.
Shear forces arising from the frictional resistance of the solid boundary and the effects of viscosity and turbulence act to oppose the gravitationally induced flow. Pressure forces are present if there is a downstream gradient in depth, and may act in the upstream or downstream directions, depending on the direction of the gradient. As noted above, the Coriolis and centrifugal forces are apparent forces that arise from the earth’s rotation and curvature of flow paths, respectively, when describing flows in a fixed coordinate system.
The nature of fluid pressure and shear are described further in the remainder of this section, and chapter 7 is devoted to a quantitative exploration of all forces in open-channel flows.
4.2.2.2 Fluid Pressure
Fluid pressure ([F L−2] or [M L−1T−2]), is the force normal to a surface due to the weight of the fluid above the surface, divided by the area of the surface. Like temperature, it is a state variable that may vary as a function of space and time. Pressure is a component of the potential energy of fluids (discussed more fully in section 4.5.1),
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• (a)
(b)
(c)
Figure 4.3 Streamlines in steady flows. The heavy arrows are velocity vectors at arbitrary points; streamlines are tangent to the time-averaged velocity vector at every point. Because the flows are steady, the streamlines are also time-averaged pathlines tracing the movement of fluid elements. (a) Steady nonuniform flow. Clearly, the direction and magnitude of velocity of fluid elements moving along the streamlines change spatially. (b) Steady nonuniform flow.
Although the direction in which element is moving is constant, the magnitude of velocity changes spatially. (c) Steady uniform flow. The direction and magnitude of velocity of each fluid element remain constant.
146 FLUVIAL HYDRAULICS
and spatial differences in pressure create forces that cause accelerations and affect the movement of fluid elements. Here, we develop expressions for the magnitude of pressure in open-channel flows and show that the pressure at a point in a fluid is a scalar quantity that acts equally in all directions.
Magnitude To derive an expression for the magnitude of pressure, consider a horizontal plane of area Ah at a depth h in a static (nonflowing) body of water (figure 4.4a). The weight of the water column is · h · Ah, where is the weight density of water, so the total pressure on the plane, Pabs, is
Pabs= · h · Ah/Ah+ Patm= · h + Patm, (4.12) where Patmis atmospheric pressure.
We shall see in section 4.5.1 that pressure is one component of potential energy, and in section 4.7 that flow is caused by spatial gradients in potential energy. Thus, we will almost always be concerned with pressure gradients rather than actual pressures,
h
Ah
θs
h·cosθS
h
A h Patm
Patm (a)
(b)
Figure 4.4 Definitions of terms for deriving the expression for pressure in (a) a water body at rest and (b) an open-channel flow (equation 4.13). See text.
BASIC CONCEPTS AND EQUATIONS 147 and since atmospheric pressure is essentially constant for a given situation, we can neglect Patmand be concerned only with the gage pressure, P:
P= · h = · g · h, (4.13a)
where is the mass density of water, and g is the gravitational acceleration. Because the situation in figure 4.4a is static, the pressure given by equation 4.13a is the hydrostatic pressure.
When water is flowing, the water surface is no longer horizontal but slopes at an angle S (figure 4.4b) in the direction of flow. The force of gravity acts vertically, but since the depth is measured normal to the surface, the pressure in this situation is given by
P= · h · cos S= · g · h · cos S. (4.13b) However, since natural stream slopes almost never exceed 0.1 rad (5.7◦), cos Sis almost always greater than 0.995, and can usually be assumed= 1.
Equations 4.13a and 4.13b, represent the hydrostatic pressure distribution and