Manning’s Equation

Water flows in a sloping drainage channel because of the force of gravity. Flow is resisted by the friction between the water and wetted surface of the channel. The quantity of water flowing (Q), the depth of flow (y), and the velocity of flow (V) depend upon the channel shape, roughness (n), and slope (S). Various equations have been devised to determine the velocity and discharge in open channels. A useful equation is the one that is named for Robert Manning, an Irish engineer.

Manning’s equation (HDS-4 Section 4.3.1) is valid for steady, uniform and turbulent flow, has the following form:

Q = (k/n)⋅A⋅R^2/3⋅S^½,

where:

k = 1.49 (US Customary units) or 1 (metric units),
n is Manning’s roughness coefficient,
A is the flow cross-section area (square feet or square meters),
R is the hydraulic radius (feet or meters), and
S is the longitudinal slope (feet/feet or meters/meters).

The hydraulic radius R is found by dividing the cross sectional area A by the wetted perimeter P. These cross-section properties are shown below for an open channel.

Over many decades, typical Manning’s n values have been compiled allowing an engineer to estimate the appropriate value by knowing the general nature of the channel boundaries. Most hydraulics textbooks and drainage design manuals provide tables of typical Manning’s n values. An abbreviated list of such Manning’s roughness coefficients is given in Appendix B, Table B.2 of HDS-4. Several pictorial guides are also available showing the Manning’s n value for different types of channels and floodplains (Barnes 1967 and Acrement and Schneider 1984). Special considerations exist for very steep channels (Jarrett 1985).

A numerical approach for n value estimates consists of the selection of a base roughness value for a straight, uniform, and smooth channel in the materials involved, and then adding values for the channel under consideration:

n = (n₀ + n₁ + n₂ + n₃ + n₄)•m₅

(4.2)

where:

n_o = Base value for straight uniform channels
n₁ = Additive value due to cross-section irregularity
n₂ = Additive value due to variations of the channel
n₃ = Additive value due to obstructions
n₄ = Additive value due to vegetation
m₅ = Multiplication factor due to sinuosity

A discussion of this method and coefficients can be found in Cowan (1956) and Chow (1959). This method may be useful for natural channels, but has limited application for most roadway drainage design work.

For rock riprap channels the Manning’s n is often described as some function of the rock size. Several equations are provided in HEC-15.

Roughness characteristics on the floodplain are complicated by the presence of vegetation, natural and artificial irregularities, buildings, undefined direction of flow, varying slopes, and other complexities. Resistance factors reflecting these effects must be selected largely on the basis of past experience with similar conditions. In general, resistance to flow is large on the floodplains. In some instances, conditions are further complicated by deposition of sediment and development of dunes and bars which affect resistance to flow and direction of flow.

The presence of ice affects channel roughness and resistance to flow in various ways. When an ice cover occurs, the open channel is more nearly comparable to a closed conduit. There is an added shear stress developed between the flowing water and ice cover. This surface shear is much larger than the normal shear stresses developed at the air-water interface. The ice-water interface is not always smooth. In many instances, the underside of the ice is deformed so that it resembles ripples or dunes observed on the bed of sand-bed channels. This may cause overall resistance to flow in the channel to be further increased. With total or partial ice cover, the drag of ice retards flow, decreasing the average velocity and increasing the depth.

When a channel cross section is irregular in shape such as one with a relatively narrow deep main channel and wide shallow overbank area, the cross section must be subdivided and the flow computed separately for the main channel and overbank area. The same procedure is used when different parts of the cross section have different roughness coefficients. In computing the hydraulic radius of the subsections, the water depth common to the two adjacent subsections is not counted as wetted perimeter (see Example Problem 4.3).

Conveyance can be computed and a curve drawn for any channel cross section. The area and hydraulic radius are computed for various assumed depths and the corresponding value of K is computed from the equation. Values of conveyance are plotted against the depths of flow and a smooth curve connecting the plotted points is the conveyance curve. If the section was subdivided, the conveyance of each subsection (Ka, Kb,…Kn) is computed and the total conveyance of the channel is the sum of the conveyances of the subsections. Discharge can then be computed using Equation 4.7

Example Problem 4.3 illustrates a conveyance curve for a compound cross section. The concept of channel conveyance is useful when computing the distribution of overbank flood flows in the stream cross section and the distribution through the openings in a proposed stream crossing. The discharge through each opening can be assumed to have the same ratio to the total discharge as the ratio of conveyance of the opening bears to the total conveyance of the channel.

4.3.2 Aids in the Solution of Manning’s Equation

Equations for the computation of Area, A, wetted perimeter, P, and hydraulic radius, R, in rectangular and trapezoidal channels (Figure 4.2) are:

A = B•y + Z•y²	(4.8)
P = B + 2•y•*√(1 + Z²)	(4.9)
R = (B•y + Z•y²)/(B + 2•y•*√(1 + Z²))	(4.10)

Variables are defined in Figure 4.2.

Figure 4.2. Trapezoidal channel.

Manning’s n For Riprap-Lined Channels

HEC-15 Section 6.1 – MANNING’S ROUGHNESS

Manning’s roughness is a key parameter needed for determining the relationships between depth, velocity, and slope in a channel. However, for gravel and riprap linings, roughness has been shown to be a function of a variety of factors including flow depth, D₅₀, D₈₄, and friction slope, S_f. A partial list of roughness relationships includes Blodgett (1986a), Limerinos (1970), Anderson, et al. (1970), USACE (1994), Bathurst (1985), and Jarrett (1984). For the conditions encountered in roadside and other small channels, the relationships of Blodgett and Bathurst are adopted for this manual.

Blodgett (1986a) proposed a relationship for Manning’s roughness coefficient, n, that is a function of the flow depth and the relative flow depth (d_a/D₅₀) as follows (Equation 6.1):

n = α⋅d_a^1/6/(2.25 + 5.23⋅log(d_a/D₅₀))

(6.1)

where,

n = Manning’s roughness coefficient, dimensionless
d_a = average flow depth in the channel, m (ft)
D₅₀ = median riprap/gravel size, m (ft)
α = unit conversion constant, 0.319 (SI) and 0.262 (CU)

Equation 6.1 is applicable for the range of conditions where 1.5 ≤ d_a/D₅₀ ≤ 185. For small channel applications, relative flow depth should not exceed the upper end of this range.

Some channels may experience conditions below the lower end of this range where protrusion of individual riprap elements into the flow field significantly changes the roughness relationship. This condition may be experienced on steep channels, but also occurs on moderate slopes. The relationship described by Bathurst (1991) addresses these conditions and can be written as follows (See Appendix D for the original form of the equation):

n = α⋅d_a^1/6 / (√g⋅f(Fr)⋅f(REG)⋅f(CG))

(6.2)

where,

d_a = average flow depth in the channel, m (ft)
g = acceleration due to gravity, 9.81 m/s² (32.2 ft/s²)
Fr = Froude number
REG = roughness element geometry
CG = channel geometry
α = unit conversion constant, 1.0 (SI) and 1.49 (CU)

Equation 6.2 is a semi-empirical relationship applicable for the range of conditions where 0.3<d_a/D₅₀<8.0. The three terms in the denominator represent functions of Froude number, roughness element geometry, and channel geometry given by the following equations:

	f(Fr) = (0.28⋅Fr/b)^log(0.755/b)	(6.3)
	f(REG) =13.434⋅(T/D₅₀)^0.492b^{1.025⋅(T/D₅₀)^0.118}	(6.4)
	f(CG) = (T/d_a)^-b	(6.5)

where,

T = channel top width, m (ft)
b = parameter describing the effective roughness concentration.

The parameter b describes the relationship between effective roughness concentration and relative submergence of the roughness bed. This relationship is given by:

b = 1.14⋅(D₅₀/T)^0.453(d_a/D₅₀)^0.814

(6.6)

Equations 6.1 and 6.2 both apply in the overlapping range of 1.5 ≤ d_a/D₅₀ ≤ 8. For consistency and ease of application over the widest range of potential design situations, use of the Blodgett equation (6.1) is recommended when 1.5 ≤ d_a/D₅₀. The Bathurst equation (6.2) is recommended for 0.3<d_a/D₅₀<1.5.

As a practical problem, both Equations 6.1 and 6.2 require depth to estimate n while n is needed to determine depth setting up an iterative process.

Trapezoidal Channel: Manning’s n

Caltrans Highway Design Manual

Commonly accepted values for Manning’s roughness coefficient are provided in Table 866.3A. The tabulated values take into account deterioration of the channel lining surface, distortion of the grade line due to unequal settlement, construction joints and normal surface irregularities. These average values should be modified to satisfy any foreseeable abnormal conditions (Reference: Caltrans Highway Design Manual Index 866.3(3)).

**Table 866.3A Average Values for Manning’s n**
Type of Channel		n value
Unlined Channels:
	Clay Loam	0.023
	Sand	0.020
	Gravel	0.030
	Rock	0.040
Lined Channels:
	Portland Cement Concrete	0.014
	Sand	0.020
	Gravel	0.030
	Rock	0.040
Lined Channels:
	Portland Cement Concrete	0.014
	Air Blown Mortar (troweled)	0.012
	Air Blown Mortar (untroweled)	0.016
	Air Blown Mortar (roughened)	0.025
	Asphalt Concrete	0.016 – 0.018
	Sacked Concrete	0.025
Pavement and Gutters:
	Portland Cement Concrete	0.013 – 0.015
	Hot Mix Asphalt Concrete	0.016 – 0.018
Depressed Medians:
	Earth (without growth)	0.016 – 0.025
	Earth (with growth)	0.05
	Gravel (d₅₀ = 1 in. flow depth < 6 in.)	0.040
	Gravel (d₅₀ = 2 in. flow depth < 6 in.)	0.056
NOTES:
	For additional values of n, see HEC No. 15, Tables 2.1 and 2.2, and “Introduction to Highway Hydraulics”, Hydraulic Design Series No. 4, FHWA Table 14. (No such table. Table B-2 provides n values.)

HEC-15

Section 2.1.3 Resistance to Flow

For rigid channel lining types, Manning’s roughness coefficient, n, is approximately constant. However, for very shallow flows the roughness coefficient will increase slightly. (Very shallow is defined where the height of the roughness is about one-tenth of the flow depth or more.)

For a riprap lining, the flow depth in small channels may be only a few times greater than the diameter of the mean riprap size. In this case, use of a constant n value is not acceptable and consideration of the shallow flow depth should be made by using a higher n value.

Tables 2.1 and 2.2 provide typical examples of n values of various lining materials. Table 2.1 summarizes linings for which the n value is dependent on flow depth as well as the specific properties of the material. Values for rolled erosion control products (RECPs) are presented to give a rough estimate of roughness for the three different classes of products. Although there is a wide range of RECPs available, jute net, curled wood mat, and synthetic mat are examples of open-weave textiles, erosion control blankets, and turf reinforcement mats, respectively. Chapter 5 contains more detail on roughness for RECPs.

Table 2.2 presents typical values for the stone linings: riprap, cobbles, and gravels. These are highly depth-dependent for roadside channel applications. More in-depth lining-specific information on roughness is provided in Chapter 6. Roughness guidance for vegetative and gabion mattress linings is in Chapters 4 and 7, respectively.

**Table 2.1. Typical Roughness Coefficients for Selected Linings**
		Manning’s n¹
Lining Category²	Lining Type	Maximum	Typical	Minimum
Rigid	Concrete	0.015	0.013	0.011
	Grouted Riprap	0.040	0.030	0.028
	Stone Masonry	0.042	0.032	0.030
	Soil Cement	0.025	0.022	0.020
	Asphalt	0.018	0.016	0.016
Unlined	Bare Soil	0.025	0.020	0.016
Unlined	Rock Cut (smooth, uniform)	0.045	0.035	0.025
RECP	Open-weave textile	0.028	0.025	0.022
	Erosion control blankets	0.045	0.035	0.028
	Turf reinforement mat	0.036	0.030	0.024
¹Based on data from Kouwen, et al. (1980), Cox, et al. (1970), McWhorter, et al. (1968) and Thibodeaux (1968). ²Minimum value accounts for grain roughness. Typical and maximum values incorporate varying degrees of form roughness.

**Table 2.2. Typical Roughness Coefficients for Riprap, Cobble, and Gravel Linings**
		Manning’s n for Selected Flow Depths¹
Lining Category	Lining Type	0.15 m (0.5 ft)	0.50 m (1.6 ft)	1.0 m (3.3 ft)
Gravel Mulch	D₅₀ = 25 mm (1 in.)	0.040	0.033	0.031
Gravel Mulch	D₅₀ = 50 mm (2 in.)	0.056	0.042	0.038
Cobbles	D₅₀ = 0.1 m (0.33 ft)	–²	0.055	0.047
Rock Riprap	D₅₀ = 0.15 m (0.5 ft)	–²	0.069	0.056
Rock Riprap	D₅₀ = 0.1 m (0.33 ft)	–²	–²	0.080
¹Based on Equation 6.1 (Blodgett and McConaughy, 1985). Manning’s n estimated assuming a trapezoidal channel with 1:3 side slopes and 0.6 m (2 ft) bottom width. ²Shallow relative depth (average depth to D₅₀ ratio less than 1.5) requires use of Equation 6.2 (Bathurst, et al., 1981) and is slope-dependent. See Section 6.1.

Circular Pipe: Manning’s n

Caltrans HDM Table 851.2

Suggested values for Manning’s Roughness coefficient (n) for design purposes are given in the tabe below.

Manning “n” Value for Alternative Pipe Materials⁽¹⁾

Type of Conduit

Recommended Design Value

“n” Value Range

Corrugated Metal Pipe⁽²⁾

(Annular and Helical)⁽³⁾

2⅔” x ½”

corrugation

0.025

0.022 – 0.027

3″ x 1″

“

0.028

0.027 – 0.028

5″ x 1″

“

0.026

0.025 – 0.026

6″ x 2″

“

0.035

0.033 – 0.035

9″ x 2½”

“

0.035

0.033 – 0.037

Concrete Pipe

Pre-cast

0.012

0.011 – 0.017

Cast-in-place

0.013

0.012 – 0.017

Concrete Box

0.013

0.012 – 0.018

Plastic Pipe (HDPE and PVC)

Smooth Interior

0.012

0.010 – 0.013

Corrugated Interior

0.022

0.020 – 0.025

Spiral Rib Metal Pipe

¾” (W) x 1″ (D) @ 11½” o/c

0.013

0.011 – 0.015

¾” (W) x Ÿ” (D) @ 7½” o/c

0.013

0.012 – 0.015

¾” (W) x 1″ (D) @ 8½” o/c

0.013

0.012 – 0.015

Composite Steel Spiral Rib Pipe

0.012

0.011 – 0.015

Steel Pipe, Ungalvanized

0.015

—

Cast Iron Pipe

0.015

—

Clay Sewer Pipe

0.013

—

Polymer Concrete Grated Line Drain

0.011

0.010 – 0.013

Notes:
(1)	Tabulated n-values apply to circular pipes flowing full except for the grated line drain. See Note 5.
(2)	For lined corrugated metal pipe, a composite roughness coefficient may be computed using the procedures outlined in the HDS No. 5, Hydraulic Design of Highway Culverts.
(3)	Lower n-values may be possible for helical pipe under specific flow conditions (refer to FHWA’s publication Hydraulic Flow Resistance Factors for Corrugated Metal Conduits), but in general, it is recommended that the tabulated n-value be used for both annular and helical corrugated pipes.
(4)	For culverts operating under inlet control, barrel roughness does not impact the headwater. For culverts operating under outlet control barrel roughness is a significant factor. See Index 825.2 Culvert Flow.
(5)	Grated Line Drain details are shown in Standard Plan D98C and described under Index 837.2(6) Grated Line Drains. This type of inlet can be used as an alternative at the locations described under Index 837.2(5) Slotted Drains. The carrying capacity is less than 18-inch slotted (pipe) drains.

HDS-4 Table B-3

**Manning’s n Values for Closed Conduits**
Description		Manning’s n Range
Concrete pipe		0.011 – 0.013
Corrugated metal pipe or pipe-arch:
Corrugated Metal Pipes and Boxes, Annular or Helical Pipe (Manning’s n varies with barrel size)	68 by 13 mm (2⅔ x ½ in.) corrugations	0.022 – 0.027
	150 by 25 mm (6 x 1 in.) corrugations	0.022 – 0.025
	125 by 25 mm (5 x 1in.) corrugations	0.025 – 0.026
	75 by 25 mm (3 x 1 in) corrugations	0.027 – 0.028
	150 by 50 mm (6 x 2 in.) structural plate corrugations	0.033 – 0.035
	230 by 64 mm (9 x 2-1/2 in.) structural plate corrugations	0.033 – 0.037
Corrugated Metal Pipes Helical Corrugations, Full Circular Flow	68 by 13 mm (2⅔ x ½ in.) corrugations	0.012 – 0.024
Spiral Rib Metal Pipe	Smooth walls	0.012 – 0.013
Vitrified clay pipe		0.012 – 0.014
Cast-iron pipe, uncoated		0.013
Steel pipe		0.009 – 0.013
Brick		0.014 – 0.017
Monolithic concrete:
1. Wood forms, rough		0.015 – 0.017
2. Wood forms, smooth		0.012 – 0.014
3. Steel forms		0.012 – 0.013
Cemented rubble masonry walls:
1. Concrete floor and top		0.017 – 0.022
2. Natural floor		0.019 – 0.025
Laminated treated wood		0.015 – 0.017
Vitrified clay liner plates		0.015
NOTE: The values indicated in this table are recommended Manning’s n design values. Actual field values for older existing pipelines may vary depending on the effects of abrasion, corrosion, deflection, and joint conditions. Concrete pipe with poor joints and deteriorated walls may have n values of 0.014 to 0.018. Corrugated metal pipe with joint and wall problems may also have higher n values, and in addition, may experience shape changes which could adversely effect the general hydraulic characteristics of the pipeline.

Other: Variation of n with Flow Depth in Pipe

From “Scattergraph’s Principles and Practice, by Kevin L Enfinger, P.E. and James S Schutsbach, ADS Environmental Services, 2003.

A fourth order polynomial approximation of Camp’s varying roughness coefficient:

f(d)=1.04+2.30*(d/D)-6.86*(d/D)²+7.79*(d/D)³-3.27*(d/D)⁴

From http://www.engineeringexceltemplates.com, Manning Equation Partially Filled Circular Pipes:

The Manning equation was developed for flow in open channels with rectangular, trapezoidal, and similar cross-sections. It works very well for those applications using a constant value for the Manning roughness coefficient, n. Better agreement with experimental measurements is obtained for partially full pipe flow, however, by using the variation in Manning roughness coefficient developed by Camp …

The equations to calculate n/_nfull, in terms of (y/D) for y < (D/2) are as follows:>/p>

n/n_full = 1 + (y/D)*(1/3) for 0 < (y/D) < 0.03
n/n_full = 1.1 + ((y/D) – 0.03)*(12/7) for 0.03 < y/D < 0.1
n/n_full = 1.22 + ((y/D) – 0.1)*(0.6) for 0.1 < (y/D) < 0.2
n/n_full = 1.29 for 0.2 < (y/D) < 0.3
n/n_full = 1.29 – ((y/D) – 0.3)*(0.2) for 0.3 < (y/D) < 0.5

The equation used for n/n_full for 0.5 < (y/D) < 1 is:

n/n_full = 1.25 – [((y/D) – 0.5)/2]