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Solving for Streamflow Without Using Manning’s Equation

by John F. Orsborn, PE, and Jeanne Stypula, PE 

In the April 2000 issue of Stream Notes Douglas J. Trieste asked, “Can no better equation than the Manning equation be developed, or are the concepts and principles used in 1889 still the best today?” (Trieste, 2000).  The answers are yes and yes; we just have to apply the principles using different concepts. 

In 1768, Chezy completed his calculations for the channel of the new Paris water supply.  Chezy was the first to consider the wetted perimeter of channels as an analog of boundary resistance (Rouse and Ince, 1957).  Chezy’s equation has the form of V = C R^1/2 S_c^1/2  that uses a resistance coefficient C, the hydraulic radius (R=A/P) and the slope of the channel, S_c.  In English units, Chezy’s C = (1.49/n)R^1/6 in Manning’s equation, where n is a resistance factor (Roberson, Cassidy and Chaudhry, 1988).  Manning’s n, when used as a roughness coefficient, should be applied to channels with a uniform surface roughness such as concrete.

The Manning equation requires the estimation of a resistance coefficient n when sizing a channel or estimating a stream flow.  There are two ways to avoid using n.  The first is to make sufficient discharge measurements at a site so that n can be measured as a function of Q, then written in terms of Q and substituted into Manning’s equation.  The resistance coefficient, n, varies inversely with Q, so n = (x/(Q)^y) where x and y are the coefficient and exponent determined from the data for n versus Q.  But, these calculations also require the measurement of the water surface slope and the calculation of the energy gradient to determine Manning's n.  The best stream gaging accuracy is plus or minus 5%.

The second way to eliminate Manning’s n is to use two equations.  One equation comes from the hydraulic geometry relationships.  The other equation describes the shear-shape relations developed by Stypula (1986) and applied to some Oregon coastal streams by Orsborn and Stypula (1987).  Usually the traditional analysis of hydraulic geometry is applied to streams based on the continuity equation:  Q = AV = WDV; and W = aQ^b, D = cQ^d and V = eQ^f where W is the water surface width;  D the mean depth;  and V the mean velocity (Leopold and Maddock, 1953).  We know from Chezy’s and Manning’s works that V is a function of the hydraulic radius (R), which is the flow area (A) divided by the wetted perimeter (P).  Including these two factors in the suite of hydraulic geometry equations, we have A = gQ^h and P = iQ^j .  Wetted perimeter accounts for two influences; the resistance to flow (shear), and a measure of available habitat for certain life-stages of fish.

Stypula (1986) developed the dimensionless relationships shown in Figure 1 for rectangular channels and natural channels from the sources listed in the legend.  The streams are located in Washington, Idaho, Montana, Wyoming, Arizona, Nevada, Vermont, and Alaska.  The widths (W) ranged from over 1,500 feet in the Columbia River to 2 inches for erosion rills in the Palouse Hills.  Mean depths (D) ranged from 27 feet to 2 inches.  The curves can be calculated at any level of flow above which the flow is in a single channel, or for a dry channel.  As shown in Figure 1, the natural, non-rectangular channels tend to follow a line defined by

                        W/D  =  P²/A - (2+2D/W)          (1)

until they approach W/D of less than 2.  Then they shift to the shear-shape line for rectangular channels defined by

                        W/D  =  P²/A - (4 + 4D/W)                    (2)

The P²/A terms give a positive number from which the factor inside the parentheses can be subtracted to compare with W/D.  The P²/A terms are the same as P/R.

Figure 1.  Shear-shape relationships for natural (real) and rectangular stream channels (Stypula 1986).

A more complete analysis of the conditions in Figure 1 is shown in Figure 2, which was developed as part of a low flow study below the dams on the lower Elwha River on the Olympic Peninsula in Washington (Orsborn and Orsborn, 1999).  Note that:  (1) the lines for non-rectangular channels and rectangular channels coincide when W/D is 40 or more (this is the basis for the “wide” channel assumption used in open channel flow calculations when R approaches D);  (2)  rectangular channels have a maximum value of A/P² at W/D = 2, which fits the assumption of hydraulic efficiency when a semicircular radius of 1/2 W just fits in a rectangular channel and

W = 2D;  and (3)  for natural, non-rectangular channels the maximum value of A/P² occurs at W/D = 1.5, and beyond that point, all natural channels coincide with rectangular channels because the wetted perimeter (P) dominates the geometric relationships.  These narrow channels are hydraulically inefficient, but are good for upstream fish passage  (Denil 1937; Ziemer 1962).

To combine the hydraulic geometry with the shear-shape equation, one must merely substitute

W = a(Q)^b   (or D = cQ^d) into Eqs. (1) or (2), which gives

                        aQ^b /D  =  P² /A - (2 + 2D/W)     (1a)

for natural, non-rectangular channels, and

                        W/cQ^d  =  P²/A - (4 + 4D/W)      (2a)

for both natural and artificial rectangular channels. And, Manning’s resistance coefficient has been eliminated.

Figure 2.  Comprehensive shear-shape relationships (Orsborn and Orsborn 1999).

The variable -(4 + 4D/W) for rectangular channels is easily determined by assigning a depth of 1 ft and various bottom widths from 1 to say 40 ft.  For W = 1 ft and D = 1 ft, W/D = 1,

P = 3 ft, P² = 9 ft², A = 1 ft² and P²/A = 9.  The correction variable is -(4 + 4(1)/1) = 8, and

W/D = 1.  The natural channel variable, -(2 + 2D/W), was determined for average conditions in numerous cross sections.  If a natural, non-rectangular channel does not fit Eq. 1a, it is probably out of balance (not at grade).  For narrower channels, the wetted perimeter (side resistance) becomes dominant and all channels follow the rectangular channel graph in Figure 2.  The correction factor -(4 + 4D/W) still holds.

For a planning level investigation at an ungaged stream site, the hydraulic geometry can be measured at a low and an intermediate flow, and graphically projected to bankfull flow.  A detailed cross-section at the site will provide the channel geometric characteristics to combine with the two flow measurements and complete the analysis.

Conclusion

We have developed equations for streamflow without Manning's n by combining channel hydraulic geometry and a dimensionless shear-shape relationship of channel characteristics.  We have verified the relationships for some Oregon coastal streams (Orsborn and Stypula 1987) and elsewhere in the states covered in Figure 1.  The Oregon results given in Table 1 demonstrate some of the benefits of using a morphological approach to find Q.

People have been using Manning's n for such a long time it may be difficult for some to embrace the concept that shape controls flow.  Shape is the result of bed and bank material and slope.

Table 1.  Measured and estimated values of average annual flow, width, depth and velocity for Deer, Fall and Flynn Creeks in the Oregon mid-coast region (Orsborn and Stypula 1987).

NOTES:

                              ^a Assumes P = W + 2D, rectangular section.

                              ^b Assumes P = W + D in natural channels, and P = W for Flynn Creek.

                              ^c Actual sizes are based on hydraulic geometry at the gaging stations.

                              ^d Est. from equations for W, D and V based on Q_a of record at 10 Regional USGS gaging stations.

References

Barnes, H. H.  1967.  Roughness characteristics of natural channels.  Water-Supply Paper 1849.  U. S. Geological Survey.

Chrostowski, H. P.  1972.  Stream habitat studies on the Uinta and Ashley National Forests.  Forest Service Intermountain Region, USDA, Central Utah Project, Ogden, UT.

Copp, H. C. and Rundquist, J. N.  1977.  Hydraulic characteristics of the Yakima River for anadromous fish spawning.  Technical Report HY-2/77.  Albrook Hydraulics Laboratory, Washington State University, Pullman, WA.

Denil, G.  1937.  La mechanique du possion de riviere.  Chap. X.  Les capacities mechaniques de la truite et du saumon.  Ann. Trav. 38(3):411-433, Belgium.

Emmett, W. W.  1975.  The channels and waters of the upper Salmon River area, Idaho.  Prof. Paper 870-A.  U. S. Geological Survey.

Leopold, L. B. and Maddock, T., Jr.  1953.  The hydraulic geometry of stream channels and some physiographic implications.  Prof. Paper 252.  U. S. Geological Survey. 

Orsborn, J. F. and J. M. Stypula.  1987.  New models of hydrological and stream channel relationships.  Erosion and Sedimentation, Pacific Rim.  Corvallis, OR.

Orsborn, J. F. and M. T. Orsborn.  1999.  Low flow assessment of the Lower Elwha River.   Elwha Tribal Fisheries.  Port Angeles, WA.

Roberson, J. A., J. J. Cassidy and M. H. Chaudhry.  1988.  Hydraulic Engineering.  Houghton Mifflin Company.  Boston, MA.

Rouse, H and S. Ince. 1957.  History of Hydraulics.  Iowa Institute of Hydraulic Research, State Univ. of Iowa, Iowa City, IA.

Stypula, J. M.  1986.  An investigation of several streamflow and channel form relationships.  M.S. thesis.  Department of Civil and Environmental Engineering, Washington State University.  Pullman, WA.

Trieste, D. J.  2000.  Manning’s equation and the internal combustion engine.  Stream Notes.  Stream Systems Technology Center.  RMRS.  Ft. Collins, CO.

U. S. Geological Survey.  1974-1980.  Measurement summary sheets, Susitna R. near Gold Creek, AK.  Gaging Sta. No. 15292000.

Ziemer, G. L. 1962.  Steeppass fishway development.  Alaska Department of Fish and Game Informational Leaflet No. 12.  Subport Building, Juneau, AK.

John F. Orsborn is Professor Emeritus of Civil and Environmental Engineering from Washington State University, and a consulting engineer in Pert Ludlow, WA;  (360) 437-0670; orsborn@olympus.net.

Jeanne M. Stypula is a Senior Engineer in the Rivers Section, Water and Land Resources Division, King County Department of Natural Resources, Seattle, WA; (206) 296-8380; jeanne.stypula@metrokc.gov.\