Physics/Essays/Anonymous/Flow in a curb or gutter system

• See this explanation from the University of Texas

Ideal open channel flow can be observed via flow in a curb/gutter system. Such a system produces uniform flow that occurs at predictable times and provides easy access for measuring. The curb/gutter system cross section shape can be idealized as a triangular cross section. This project provides the details of a field exploration of flow in a curb/gutter system.

Observed Cross Section Details

The observed curb/gutter system is located on Hearthstone Drive in w:Blacksburg, Virginia. Located in an apartment complex parking lot, this system provides a segment of channel with a few imperfections. The segment totals 37 feet in length, includes a slight slope of 4.43%, and provides conveyance from a parking lot in a nearby storm drain. The following figures display the observed curb and gutter system, and a cross section of the curb and gutter.

Computerized model of a surveyed curb and gutter system.

Theoretical Rating Curve

A rating curve displays the relationship between discharge and depth. The depth, typically referred to as stage, is measured from the very bottom of the channel to the flow's surface. The magnitude of flow, or discharge (Q), is dependent upon stage and channel shape and can be found using Manning's equation. Manning's equation computes velocity as a function of channel slope, hydraulic radius, Manning's n coefficient, and a conversion factor. Manning's equation is given as
$v={k \over n}R^{2/3}S^{1/2}$
where:
v = velocity of flow (length/time)
k = conversion factor equal to 1 for metric units or 1.49 for English units
n = Manning's n coefficient
R = hydraulic radius = channel cross sectional area/wetted perimeter (length)
S = friction slope of channel (length/length)

Flowrate can also be estimated using Manning's equation by applying the discharge formula.
$Q=A\cdot v$
where:
Q = flowrate (length³/time)
A = cross-sectional area of flow (length²)

Therefore, flow can be estimated by
$Q={k \over n}AR^{2/3}S^{1/2}$

For this application, the cross section of the curb can be simplified to form a triangular channel.

Idealized curb and gutter system into a triangular cross section.

Thus the channel cross sectional area may be computed with the following.
$A={1 \over 2}By={1 \over 2}my^{2}$
where:
B = top width of the flow = m \cdot y (length)
y = depth of flow (length)
m = slope of the triangle (length/length)

When channel shape is known, a theoretical rating curve can be developed by solving Manning's equation for varying depths.

Theoretical rating curve for chosen curb and gutter cross section.

Manning's n

Manning's n, denoted as n, quantifies a channel's roughness. In any flow scenario, channel roughness can play a large role in the rate at which water conveys through the channel. A higher value of Manning's n indicates that a channel is rough and thereby creates more friction as the water moves through the channel. This friction causes a loss in energy and consequentially a reduction in flow velocity. A smaller n value has the opposite effect. The Manning's n value for common materials, such wood or cement can be found in literature. The observed channel material is Portland cement concrete. For the purpose of this exploration a Manning's n of 0.013, the value associated with float finish concrete, was used.

Classification of Flow

Open channel flow can be characterized by a variety of terms. Flow can be considered critical, subcritical, or supercritical. Supercritical flow describes flow that is shallow and fast-moving, whereas subcritical flow describes flow that is deep and slow-moving. Critical flow describes flow located at the interface between super- and subcritical flow. The Froude number, a dimensionless number that provides a ratio between flow velocity and shallow wave velocity, can be utilized to determine flow classification. The formula for the Froude number is
$F_{r}={v \over {\sqrt {g{A \over B}}}}$
where:
F_r = Froude number g = acceleration of gravity (length/time²)

When the Froude number is below 1, flow is subcritical. When the Froude number is above 1, flow is supercritical. When the Froude number equals 1, flow is critical. The theoretical Froude number(s) for the observed curb/gutter system channel were found for each theoretical velocity/flow computed, and a graph relating the two variables was created.

Theoretical Froude number versus stage plot

Depth and Reach Classification

Supercritical, subcritical, and critical depths are characterized in the same manner as supercritical, subcritical, and critical flow. However, barring critical depth, each type of depth applies over a range of depths that may be considered supercritical or subcritical. In the case of critical depth, the depth given refers to the depth at the transition between super- and subcritical depths.

Using the simplified cross section (Figure 3), the depth at critical depth can be derived from the specific energy equation:
$E={v^{2} \over 2g}+y={Q^{2} \over 2gA^{2}}+y$
where:
E = specific energy of flow (length)

In terms of only flow depth, discharge, side slope (m), and acceleration of gravity:
$E={Q^{2} \over {2g({\tfrac {1}{2}}my^{2})^{2}}}+y={2Q^{2} \over gm^{2}y^{4}}+y$
Taking the derivative and setting it equal to 0, the critical depth can be found:
${\operatorname {d} \!E \over \operatorname {d} \!y}=0=({2Q^{2} \over gm^{2}}){-4 \over y_{c}^{5}}+1$
$y_{c}=({8Q^{2} \over gm^{2}})^{1}/5$

Each reach also possesses a normal depth or the depth at which the reach flows under uniform conditions. Normal depth, y₀, is computed using the modified Manning's equation and measured Q. Knowledge of the normal depth magnitude allows for reach classification. A reach can be classified as either steep or mild. A “mild” reach has a normal depth that is greater than its critical depth whereas a “steep” reach has a normal depth that is less than its critical depth.

For the purpose of this exploration, the y₀ and y_c will be found using an average velocity taken from the collection of field data during three separate rainfall events.

Field Data Collection and Analysis

Field data collection was performed in two phases; phase I, which included the initial locating and surveying of the observed segment, and phase II, which included three separate observations of the selected curb/gutter system during periods of rainfall. Phase I yielded the aforementioned channel cross section, length, and slope through the use of surveying equipment.

Channel selection was based upon ease of access, safety, and degree of imperfections. For the surveying process a Total Station Instrument (TSI) was utilized. The TSI was placed near the curb and gutter section and the location set to a reference point of (0, 0, 0). A prism rod was used to record the northing, easting, and elevation of various locations along the segment channel bottom, effectively recording the channel slope and cross section.

Phase II occurred over the course of 3 weeks during three separate rainfall events. The desired data were obtained with a flow velocity meter and a measuring rod capable of measuring millimeters. The observed velocity and depths at three locations along the slope were collected during each of the rain events.

	Date (mm/dd/yy)	Time	Observed Velocity (at location 1)	Depth at location 1	Depth at location 2	Depth at location 3
Event 1	Example	Example	Example	Example	Example	Example
Event 2	Example	Example	Example	Example	Example	Example
Event 3	Example	Example	Example	Example	Example	Example

With the above information obtained, yo and yc could be determined. The channel velocity was found by taking the average of the three recorded event velocities and utilizing Manning's equation; resulting in a channel velocity of ____ ft/s. Through the use of Equations 2, 3, 4, and 8 it was found that yc equals ___ and yo equals___. Thus the observed curb/gutter reach was classified as ____.

The velocities for each event were transformed to discharges. These discharges and their associated measured depths were then placed on the theoretical ratings curve.

(insert figure here)

Finally, the relative error between the theoretical and observed data points was found. The formula for relative error is:
${\text{Relative Error}}={x_{t}-x_{0} \over x_{0}}\times 100\%$
where:
x_t = theoretical value
x₀ = observed value

The relative errors between the theoretical and observed discharge were computed.

	Stage (ft)	Theoretical Discharge (ft3/s)	Actual Discharge (ft3/s)	Relative Error (%)
Event 1	Example	Example	Example	Example
Event 2	Example	Example	Example	Example
Event 3	Example	Example	Example	Example
Avg. Velocity	Example	Example	Example	Example

Explanation of Error

The average relative error from the table above was found to be ___%. Section to be completed.

Conclusion

Overall, the exploration of the curb/gutter system located on Hearthstone Drive in Blacksburg, Virginia provided a useful exploration of an open channel system. When compared, the theoretical and observed computations maintained minimal relative error values. The curb.gutter system provided an excellent platformfor the confirmation of theoretical concepts.