Determining roughness coefficient (n value)

The roughness coefficient, an empirical factor in Manning’s equation, accounts for frictional resistance of the flow boundary. Estimating this flow resistance is not a simple matter. This parameter is used in computation of water surface profiles and estimation of normal depths and velocities.

Boundary friction factors must be chosen carefully, as hydraulic calculations are significantly influenced by the n choice. Factors affecting roughness include ground surface composition, vegetation, channel irregularity, channel alignment, aggradation or scouring, obstructions, size and shape of channel, stage and discharge, seasonal change, and sediment transport.

Significant guidance exists in the literature regarding roughness estimation. Chow (1959) discusses four general approaches for roughness determination. The U.S. Geological Survey (USGS) (Arcement and Schneider 1990) published an extensive step-by-step guide for determination of n values. NRCS guidance for channel n value determination is available from Faskin (1963). Finally, when observed flow data and stages are known, manual calculations or a computer program such as HEC–RAS may be used to determine n values.

With the many factors that impact roughness, and each stream combining different factors to different extents, no standard formula is available for use with measured information. As stated in Chow (1959):

…there is no exact method of selecting the n value. At the present stage of knowledge [1959], to select a value of n actually means to estimate the resistance to flow in a given channel, which is really a matter of intangibles. To veteran engineers, this means the exercise of sound engineering judgment and experience; for beginners, it can be no more than a guess, and different individuals will obtain different results.

While there has been considerable research on estimating roughness coefficients since 1959, flood plain and channel n values are still challenging to determine. In practice, to a large extent the selection of Manning’s n values remains judgement based.

Estimates of channel roughness may be made using photographs or tables provided by Chow (1959), Brater and King (1976), Faskin (1963), and Barnes (1967). NEH–5 supplement B, Hydraulics, can also be used to estimate roughness values. As roughness can change dramatically between surfaces within the same cross section, such as between channel and overbanks, a determination of a composite value for the cross section is necessary (Chow 1959). The choice of a channel compositing method is very important in stream restoration design where large differences exist in bank and bed roughness. While the following example uses the Lotter method, other methods, such as the equal velocity method and the conveyance method, can also be used.

Example problem: Composite Manning’s n value

Problem: Determine a composite n value for the cross section illustrated in figure 5 at the given depth of flow.

Assume that this channel is experiencing a 6,094 cubic feet per second flow, with 5,770 cubic feet per second in the main channel and the remainder on the right overbank. The mean velocity in the main channel is 2.3 feet per second and on the overbank, 0.55 foot per second. The channel slope is 0.00016, and a fairly regular profile of clay and silt is observed.

The channel is relatively straight and free of vegetation up to a stage of 10 feet. Above that level, both banks are lined with snags, shrubs, and overhanging trees. The right overbank is heavily timbered with standing trees up to 6 inches in diameter with significant forest litter. In stream work, the convention is that the left bank is on the left when looking downstream. See figures 6 and 7 where the photos are taken at a lower stage (Barnes 1967).

Solution: To determine the composite Manning’s n value, the inchannel and overbank n values must first be determined.

The solution will first estimate n values using reference materials, then this solution will compare this estimate with the value calculated from Manning’s equation. Roughness estimates can be found in NEH–5, Hydraulics, supplement B by Cowan (1956). Arcement and Schneider (1990) extended this body of work. Both methods estimate a base n value for a straight, uniform, smooth channel in natural materials, then modifying values are added for channel irregularity, channel cross-sectional variation, obstructions, and vegetation. After these adjustments are totaled, an adjustment for meandering is also available.

For the channel below 10 feet, the bed material is silty clay. Arcement and Schneider (1990) show base n values for sand and gravel. For firm soil, their n value ranges from 0.025 to 0.032. Cowan (1956) shows a base n of 0.020 for earth channels. Richardson, Simons, and Lagasse (2001) shows 0.020 for alluvial silt and 0.025 for stiff clay. A reasonable assumption could be 0.024 for the channel below 10 feet of depth. For the remainder of the channel, above 10 feet of depth to top of bank at 20 feet, the effects of vegetation must be added in. The channel is then divided into three pieces: a lower channel, an upper channel, and a right overbank. Other breakdowns of this cross section are possible.

For the lower channel a base n value of 0.024 is assumed. Referring to Cowan (1956) in NEH 5, supplement B, a 0.005 can be added for minor irregularity and a 0.005 addition for a shifting cross section. This gives a total n value for the lower channel of 0.034.

For the upper channel, the area above the lower 10 feet of flow depth and excluding the right overbank, the base n value is 0.024, a minor irregularity addition of 0.005, a 0.005 addition for a shifting cross section, a minor obstruction addition of 0.010, and a medium vegetation addition of 0.020 can be selected. This gives a total n value for the upper channel of 0.064.

For the overbank, a base n (from the overbank soil) is needed. Based on site-specific observations, it was found that the soil is slightly more coarse than that of the main channel, n = 0.027. Again from NEH 5, supplement B, Cowan (1956) a minor irregularity addition of 0.005, a shifting cross section addition of 0.005, an appreciable obstruction addition of 0.020, and a high vegetation addition of 0.030 can be selected. This gives a total n value for the overbank of 0.087.

To obtain composite roughness, use the method of Chow (1959), whereby a proportioning is done with wetted perimeter (P) and hydraulic radius (R):

As follows:

Using equation 21 the composite roughness is:

This value can be compared to a value calculated with Manning’s equation as follows.

Discussion:
The difference in Manning’s n initially appears to be cause for concern. However, it does illustrate three important points. First, this process is subjective, and two equally capable practitioners may arrive at different results. Second, Manning’s equation is for uniform flow. Differences in measured and calculated n values should be attributed to the uncertainty in choosing appropriate values to account for various factors associated with roughness. Manning’s equation by itself can provide an estimate, but it cannot precisely determine roughness when the flow is not uniform. Third, an uncertainty analysis is recommended for hydraulic analysis.

As documented in Barnes (1967), the USGS backwater calculations determined the channel n value to be 0.046 and the right overbank n value to be 0.097. In contrast to this example, Barnes calculated roughness using energy slope, rather than water surface slope and also included expansion and contraction losses.

Example problem: Manning’s n value for a sand-bed channel

Problem: Determine the n value for a wide, sand chan-nel with the following cross section (fig. 8). Assume a discharge of 4,100 cubic feet per second, a thalweg depth of 5 feet, 3:1 side slopes and a fairly straight, regular reach. Assume a slope of 0.0013 and a sandy bottom with a D50 of 0.3 millimeter.

Solution: Roughness in sand channels is highly dependent on the channel bedforms, and bedforms are a function of stream power and the sand gradation. Arcement and Schneider (1990) show suggested n values for various D50 values with the footnote that they apply only for upper regime flows where grain roughness is predominant. For a D50 of 0.3 millimeter, this reference suggests a 0.017 n value. However, it is important to assess the regime of the flow. A figure from Simons and Richardson (1966) (also in Richardson, Simons, and Lagasse 2001 and Arcement and Schneider 1990) is shown as figure 9. Given stream power and me-dian fall diameter, the flow regime may be estimated, as well as the expected bedform and roughness range.

Stream power may be calculated from where gamma is unit weight of water, Q is discharge, and Sf is the energy slope. Assuming the energy slope is nearly the same as the bed slope, then:

For figure 9, stream power per cross-sectional area is needed. The flow area for the given cross section is 554 ft², so the stream power is 0.60 pounds per second per square foot (per foot of channel length). Reading figure 9, with a D₅₀ of 0.3 millimeter, the flow is in the upper regime, but close to the transition. This would support an n value of 0.017, particularly if bed-forms are present.

Figure 10 (Arcement and Schneider 1990) indicates the general bedforms for increasing stream power.

The anticipated bedform is a plane bed, and figure 9 suggests an n value between 0.010 and 0.013 for plane beds. The presence of breaking waves over antidunes would raise the roughness estimate to between 0.012 to 0.02. Finally, an estimate may be calculated with the Strickler formula (Chang 1988; Chow 1959) that relates n value to grain roughness. So, for a plane bed it should give a good estimate:

or

Since the D₅₀ is 0.3 millimeter, the calculated n value is 0.012, which agrees with figure 9 results for plane beds. Arcement and Schneider (1990) show n = 0.012 for a D₅₀ of 0.2 millimeter, and this calculation is close to the transition range. Considering all of the above, information supports a roughness selection between 0.013 to 0.017. If field observations support the plane bed assumption, a value from the low end of this range should be selected. If antidunes are present, a value from the high end of this range would be reasonable.

Example problem: Manning’s n value for a gravel-bed channel

Problem: Determine the n value for a wide, gravel-bed channel with a D₅₀ of 110 millimeters. Assume a fairly straight, regular reach. Assume minimal vegetation and bedform influence.

Solution: Since the grain roughness is predominant, the Strickler formula can be used.

This results in an estimated n value of 0.033. It should be noted that this estimate does not take into account many of the factors which influence roughness in natural channels. As a result, a estimate made with Strickler’s equation is often only used as an initial, rough estimate or as a lower bound.