Transverse flow hydraulics and its geomorphologic effects
Frequently, the intent of channel design is to try to recreate or restore a natural condition, one that is geomorphologically sustainable. The hydraulic engineer needs to be aware of the mechanics of the flow and movable boundaries in channel curves. In a straight channel section, the task of determining boundary stress is easier than in curved reaches, as the direction of flow is more likely to be parallel to the banks. Shear force is dominant, and no significant additional force exists due to the momentum of flow impinging on the bank at some angle. In a curve, accounting for those angles of impinging flow is very important. The problem is three-dimensional, as previously mentioned, accounting for velocity distributions in water surface profiles, and flow in a curve sets up transverse velocity vectors and spiral motion. This phenomenon is completely natural and one of the driving mechanisms of geomorphology.
If a curving section of streambank is to be stabilized, some understanding of the nature of transverse (or secondary) flow is necessary. The task of streambank protection may be roughly divided into two major strategies: installation of measures that enable the bank to resist hydraulic forces at whatever angle they impinge or redirecting the flow so that the bank is no longer subject to damaging forces. Examples of the first would be planting vegetation on the banks or installing woody debris. The second strategy employs such measures as stream barbs, spur dikes, or longitudinal groins. However, particularly for curved channels, an examination of the hydraulic aspects upon which any streambank protection mea-sure will succeed or fail is given here.
Even in straight channels, some flow spiraling can occur, and a moveable bed sets up transverse slopes that alternate direction along the bed profile. Figure 12 (Chang 1988) illustrates the behavior of spiral flow and the resulting transverse bed slopes.
In curved sections, the secondary current is not necessarily only one cell of circulation as shown in figure 13 (Chang 1988).
Chang (1988) provides the following equation for a hydraulically rough channel:
where:
δ = angle of the bottom current with channel centerline
d = depth at the location of interest in the section
r = radius of curvature to the location of d
The channel roughness is not considered to have a significant influence on the angle δ. Chang (1988) documents research that can enable the hydraulic engineer to calculate shear stress in the radial (or transverse) direction, the transverse bed slope a channel might be expected to acquire, and the sediment sorting expected along that transverse slope.
where:
ρ = density of water
g = acceleration due to gravity
κ = the dimensionless von Kármán constant (κ ≈ 0.40)
U = avg. cross-sectional velocity
C = Chézy resistance factor, defined below
d = depth at the location of interest
r = radius of curvature to that location
f = friction factor as defined below
The Chézy resistance factor is similar to Manning’s n value in that it is an empirically derived coefficient serving as an index of boundary roughness. The following Ganguillet and Kutter formula (1869), as provided in Chow (1988), is a method of calculating Chézy C, given Kutter’s n:
where:
S = profile bed slope
R = hydraulic radius
n = Kutter’s roughness
Chézy’s C is related to Manning’s n by the following equation in English units:
The Darcy-Weisbach friction factor, f, is described by Chow (1959) and for uniform or near uniform flow may be calculated using:
Both Chow (1959) and Chang (1988) describe the relationship of f to boundary Reynolds number. Chang provides three formulas, dependent on hydraulic smoothness, for channels in which form roughness is not a factor as follows.
where:
R = hydraulic radius
Rbed= boundary Reynolds number
ks = equivalent roughness or grain roughness, calculated from the following, one of several similar equations, Chang (1988):
For the transition from hydraulically smooth to rough:
where the coefficients A0 through A6 are 1.3376, -4.3218, 19.454, -26.48, 16.509, -4.9407, and 0.57864, respectively.
For the hydraulically rough regime:
For gravel-bed rivers, Chang (1988) provides the following equation:
where:
d = max depth of flow with units same as D50
In figure 13, δ is the angle between the velocity vector of the bottom current and the centerline. Also of interest is the resultant angle of shear stress between the two components of shear, and longitudinal and radial. Chang (1988) gives that angle, δ , as:
Longitudinal shear stress at any point in the cross section is calculated with the following equation:
where the c subscript refers to the channel centerline.
The transverse bed slope (β) can be computed using:
where:
δ = the angle shown in the above sketch
ϕ = the sediment angle of repose
This equation is valid when β is small compared to ϕ. This relationship is less accurate for channels with significant quantities of suspended sediment. Since ϕ is generally >30º, then β should be less than 10º. If ϕ >30º, then β becomes less valid as δ increases toward 20º or in tight curves.
Finally, Chang (1988) provides a formula for determining sediment sorting on the transverse slope:
where:
D = median grain size
d = depth at that location
Sc = longitudinal profile slope along the centerline
rc = radius of curvature to centerline
r = radius of curvature to location of d
ρ = densities of sediment and water
Example problem: Design radius
Problem: A roughly trapezoidal curved channel is being designed with a moveable boundary in dynamic equilibrium to carry a flow of 700 cubic feet per second. The channel profile slope is 0.0013, channel bottom width is 30 feet, with a transverse bed slope, β, of 10 percent, and 3H:1V side slopes. The bed material is rounded gravel, with a D50 of 0.30 inches, and n value of 0.035. Considering uniform flow and a maximum depth of 6 feet, calculate the design radius of curvature to the centerline, longitudinal and radial stress vectors at the centerline, and the resultant stress angle in the curve.
Solution:
Part Design radius of curvature to the centerline
The angle of repose ϕ for 0.3-inch, rounded gravel is about 31 degrees. Assuming a constant transverse bed angle of 10 percent, tan β = 0.10, and the resulting angle of the bottom current would be:
or
so, δ = 9.4 degrees
Consider the channel centerline to be horizontally located at the centroid of the flow cross section, as shown in figure 14.
To find X, the flow area left of the centroid must be equated to that on the right:
Simplifying:
given:
Part 2—Longitudinal and radial stress vectors
The longitudinal shear stress at the centerline is calculated with equation 49.
The total flow area is 202.5 square feet, wetted perimeter = 58.6 feet, so R = 3.46 feet. From Q = VA, the average velocity is 700/202.5 = 3.46 feet per second. The friction factor is:
The radial shear is calculated with equations 38 and 39.
Part 3—Resultant stress angle in the curve
The direction of the resultant stress vector between the longitudinal and radial components is calculated using equations 48 and 40.
where:
Hey (1979) addresses point bar development with a sketch similar to figure 15, showing how secondary currents, along with bed-load supply, impact the location of aggradation and degradation in a meander.
During bankfull flows, the strongest velocity vectors follow the course of the arrows starting at A in figure 15, cutting across the toe of point bars with the highest bed-load supply. At B, downstream of the bar apex, the shear stress and transport capacity drop, and aggradation occurs. Opposite the point bar at C, low bed load accompanies the incoming flow, and as surface currents angle into the bank and undercurrents move away from the bank, a zone of downwelling results at point D. The low bed load gives the stream a scouring tendency. Toward the inflection point of the meander, flow with a low bed-load supply enters a contracted reach at E that is steeper and shallower, and regains its scouring capacity. Riffles form and, as the highest velocity vectors cut from one point bar toe to the toe of the next downstream bar, riffles are often skewed to the banks.