Hydro Review

Dimensionally Homogeneous Form of the Chezy and Manning Equations

Chezy and Manning equations, the foundation of our present science of open channel hydraulics, are not dimensionally homogeneous. The author presents a new derivation of these equations that reveals the constituent parts of these coefficients, allowing for calculation of more accurate values.

By Lee H. Sheldon

Open channel hydraulics was undoubtedly the first subject studied under the broad category of fluid mechanics. The Law of Continuity was the first relation to be developed. Published in 1628 by a Benedictine monk named Benedetto Castelli,1 who was a pupil of Galileo Galilei, this fluid law is formulated today as:

Equation 1
Q = AV
where:
- Q is volumetric flow rate in ft3 per second;
- A is flow cross-sectional area in ft2; and
- V is average velocity over a cross-sectional area perpendicular to the flow in feet per second.

In the 1760s, French engineer Antoine Chezy compared flow behavior between two streams having similar characteristics1 and formulated his observations as:

Equation 2
Vaverage = C(R1/2)So1/2
where:
- C is Chezy's coefficient in ft1/2 per second;
- R is hydraulic radius or area/wetted perimeter in feet; and
- So is the slope of the channel bottom (the concept of an energy slope was not yet in use).

After that, numerous investigators - including Strickler, Gauckler, Kutter, Ganguillet and Hagen - sought to devise other formulas and coefficients by finding analytical expressions that replicated observed stream flow behavior.

Finally, in 1889, Irish engineer Robert Manning postulated a formula that was an average of all the previous formulas1 and is written as:

Equation 3
Vaverage = (1.486/n)R2/3So1/2
where:
- 1.486/n is Manning's coefficient, in ft1/3/sec.

Chezy's C equals (1.486/n)R1/6 in Manning's equation. Therefore, Manning's coefficient, 1.486/n, has units of ft1/3/sec. (In the metric system, 1/n is the coefficient so that the numerical value of n is the same in both systems). It has always been known that both coefficients, C and n, are numerically variable. They appear to vary with roughness of the channel boundaries and for very shallow and very steep slopes. Further, it is unclear how seemingly similar channels could have different coefficients. Therefore, investigators had long sought to learn the component makeup of the coefficients in order to improve the accuracy of the two equations.

Modern usage

Both of these formulae are in common use today. Both equations refer to a single point of measurement along the course of the channel. Thus, they do not explicitly contain a parameter of head loss. Further their applicability has been clarified; they are for use only with normal flow. In open channel hydraulics, normal flow refers to where the slope of the channel bottom, So, slope of the water surface, Sw, and friction slope (or energy gradient or energy line), Sf, are all the same. In other words, steady state, where the hydraulic depth is not changing. In contrast, if the flow conditions are changing, they would be characterized as gradually varied flow or rapidly varied flow. The velocity profile of normal flow in an open channel is not restricted to being everywhere equal. In pipes, however, normal flow is defined as the condition where the velocity profile perpendicular to the flow cross-sectional area is everywhere equal over the area.

Velocity head

From Bernoulli's equation, the kinetic energy of a moving stream is equal to a term known as the velocity head, V2/2g. This term was originally obtained by direct integration of one of the three terms in Euler's energy equation, (1/2g)∫VdV. However, it was not until years later that Gaspard Gustave de Coriolis recognized that unless the flow is uniform (constant velocity profile), integration for an average V was not mathematically correct without including a correction, or Coriolis factor, of α.2,3,4 Thus, the actual velocity head is now correctly expressed as:

Equation 4
Hv = αV2average/2g
where:
- Hv is velocity head in feet or specific energy in ft-lbs/lb; and
- g is gravitational acceleration in ft/sec2.

Dimensional analysis

Dimensional analysis is a tool used in fluid mechanics more than in any other technical field. Due to the number of variables associated with most fluid problems, the Principle of Dimensional Homogeneity (PDH) has been developed. This states that, "If an equation expresses a proper relationship between variables in a physical process, it will be 'dimensionally homogeneous'; that is, each of its additive terms will have the same dimension."5

From the velocity head, Hv is identified as a function of Vaverage. Also, Chezy's and Manning's equations each identify Vaverage as a function of both R and So. Therefore, by equality of Vaverage, Hv can be expressed as a function of both R and So:

Equation 5
Hv = function{R, So}
In accordance with PDH, this can be reformulated as:

Equation 6
Hv/R = function{So}

In this manner, both sides of the functional equation are dimensionless. In addition, this form of the equation can be analyzed in an open-channel flume in a hydraulic laboratory. It is likely that the relation is non-linear; here it will be assumed to be a second order relation. However, the following procedure can be used equally with any order of non-linearity. As a second order, the resulting correlation equation would become:

Equation 7
Hv/R = a'So2 + b'So + c'

where:

- a', b' and c' are arbitrary constants that are dimensionless.

In this form, the equation is not of particular practical use. However, a technique has been developed in the analysis of hydropower instrumentation6 that will convert this into a more recognizable form. If the open flume test data is plotted as a log10-log10 graph, the resulting second order equation would become:

Equation 8

log10Hv/R = alog102So + blog10So + c

Dropping the subscript 10 and noting the identities of klogx = logxk and logxk + logxm = logxk + m, this can also be written:

Equation 9

logHv/R = alogSologSo + blogSo + c

= logS0alogSo+ logSob + c

= logS0alogSo + b + c

For the next steps, each term can be multiplied by log1010 = 1. The equation then becomes:

Equation 10

logHv/R = logS0alogSo + b + clog10

= logS0alogSo + b + log10c

= log10cS0alogSo + b

Finally, raising both sides of the equation as powers of 10 and using the identity 10logx = x, the equation becomes:

Equation 11

Hv/R = 10cSoalogSo + b

Now, this dimensionally homogeneous equation can be rearranged as:

Equation 12

Hv = 10cRSoalogSo + b

At this point, substitution may be made for Hv by its homogeneous equality αV2average/2g, which after rearranging becomes:

Equation 13

Vaverage = (2g10cRSoalogSo + b/α)1/2

Finally, this may be rearranged again as a new, dimensionally homologous form of Chezy's equation:

Equation 14

Vaverage = (2g10cSoalogSo + b - 1/α)1/2R1/2So1/2

Therefore, Chezy's C is now seen to be equal to:

Equation 15

C = (2g10cSoalogSo + b - 1/α)1/2

Likewise, Manning's n becomes:

Equation 16

n = 1.486α1/2 R1/6/(2g10c SoalogSo + b - 1) 1/2

Components of Chezy's C

This penultimate equation reveals that Chezy's C is:

- Proportional to g1/2, which causes the coefficient to still have dimensions but provides the units to allow this new form of Chezy's equation to become dimensionally homogeneous. This gravitational term could be separated as a standalone term and allow the coefficient to be truly dimensionless. It also shows that because of the acceleration of gravity, the coefficient, at least in theory, varies with elevation;

- Inversely proportional to α1/2. As the roughness of the channel boundaries increases, the resistance to flow becomes greater, velocity profile becomes more non-uniform and Coriolis correction factor becomes larger. This is the reason that it is known that Chezy's C and the average velocity both decrease with increasing channel rugosity (or roughness);

- Proportional to the slope, So; more specifically, it is proportional to the non-linear portion of the dimensionless relation between Hv/R and So. As noted previously, the non-linearity is not limited to a quadratic relation, but if experimentation shows a need, the method of this article allows the non-linearity to be expressed as any degree of non-linear polynomial, such as Soa1logSo + a2log^2So + a3log^3So + ... + b - 1. Further, even if the coefficients a, b, and c are experimentally determined to be truly constant, this proportionality component of Chezy's C is implicitly variable by the very nature of the exponent of the slope also containing the slope. It is for this reason that the uncertainty of Chezy's and Manning's equations are known to increase, or their accuracy decrease, as the slopes became very shallow or very steep. A comparison with present empirical values of Chezy's C shows that the non-linear slope terms do exist and are numerically significant.

Conclusion

A new, dimensionally homogeneous form of Chezy's and Manning's equations has been derived that also show the constituent components of their coefficients. Chezy's equation with its diagnosed coefficient becomes:

Equation 17

Vaverage = (2g10cSoalogSo + b - 1/α)1/2 R1/2So1/2

The coefficient of this new equation has three exponent constants from a single quadratic equation that have yet to be determined by laboratory experimentation. This new complete modified, dimensionally homogeneous, Chezy equation can also be written as:

Equation 18

Vaverage = (2g10c/α)1/2 R1/2Soa/2 logSo + b/2

Or, more generally, for an nth order polynomial, as:

Equation 19

Although the coefficients are hypothetical and not determined with experiments, one of the author's students is designing and constructing an open channel laboratory flume to measure these coefficients.

Notes

1Rouse, H., and S. Ince, History of Hydraulics, Iowa Institute of Hydraulic Research, Iowa City, Ia., 1957.

2Sheldon, Lee H., and F.J. Russell, "Determining the Net Head Available to a Turbine," Transactions of the ASME Second Symposium on Small Hydro-Power Fluid Machinery, American Society of Mechanical Engineers, New York, 1982.

3Sheldon, Lee H., "The Bernoulli Theorem: Sharing its History and Application," Hydro Review, Volume 19, No. 5, August 2000, pages 110-118.

4Sheldon, Lee H., and Rodney J. Wittinger, "Improving Turbine Efficiency Calculations through Advanced Velocity Measurements," Hydro Review, Volume 26, No. 3, June 2007, pages 52-60.

5Munson, B.R., et al., Fundamentals of Fluid Mechanics, Sixth Edition, John Wiley & Sons Inc., Hoboken, N.J., 2009.

6Sheldon, Lee H., "New Calibration Equation for the Winter-Kennedy Piezometer System," Hydro Review, Volume 36, No. 8, October 2013, pages 36-43.


This article has been evaluated and edited in accordance with reviews conducted by two or more professionals who have relevant expertise. These peer reviewers judge manuscripts for technical accuracy, usefulness, and overall importance within the hydroelectric industry.


Lee Sheldon, MSME, P.E., is a consulting engineer and professor of engineering at Oregon Institute of Technology. He is also affiliated with Black & Veatch.

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