The Effect of Bed 

Roughness on DepthDischarge Relations 

in Alluvial Channels

By D. B. SIMONS and E. V. RICHARDSON

STUDIES OF FLOW IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-E

A discussion of the variations in depthdischarge relations in alluvial channels 

and the factors that produce these 

variations

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1962
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington 25, D.C.
CONTENTS

	Page 

Abstract____--_-__---_--______________---_---__--____________ E-l

Forms of bed roughness in alluvial channels_________________________ 1

Resistance to flow in alluvial channels______________________________ 5

Effect of form of bed roughness on depth-discharge relations-___________ 6

Lower-flow-regime run with constant channel slope________________ 8

Lower- and upper-flow-regime runs______________________________ 12

Small channel slope varying with discharge___________________ 12

Moderate channel slope varying with discharge______________ 12

Large channel slope varying with discharge_-_______--__--_.__ 14

Upper- and lower-flow-regime run with channel slope varying inversely

with discharge___-__--_-_--__------_---_--_----------__- 18

Upper-flow-regime run._____________-___-_-_______-___-___-__-- 18

Other factors that influence depth- and stage-discharge relations._______ 21

Conclusions_________________ ________-__________. ____________ 24

Literature cited____-_____________________-_---_-_-___--_-__-__-__- 26

ILLUSTRATIONS

Page 

FIGUBE 1. Forms of bed roughness in alluvial-channels.________________ E-3

2. Effect of size of bed material on form of bed roughness and

Manning n________________________________________ 7

3. Variation of hydraulic variables with time for a lower- and

upper-regime-flow hydrograph.___-_-___-___-_-_--____-__ 9

4. Relation of depth to discharge for lower-regime flow when

channel slope is constant______________________________ 10

5. Relations of depth to discharge for lower- and upper-regime

flow when variation of slope with discharge is small___-___- 13

6. Relation of depth to discharge for lower- and upper-regime flow

when the variation of slope with discharge is moderate.____ 15

7. Relation of depth to discharge for lower- and upper-regime flow

when the variation of slope with discharge is large.________ 17

8. Relation of depth to discharge for upper- and lower-regime flow

when slope varies inversely with discharge________________ 19

9. Relation of depth to discharge for upper-regime flow.________ 20

m

STUDIES OF FLOW IN ALLUVIAL CHANNELS

THE EFFECT OF BED ROUGHNESS ON DEPTHDISCHARGE RELATIONS IN ALLUVIAL CHANNELS

By D. B. SIMONS and E. V. RICHAKDSON

ABSTRACT

Alluvial channel stage-discharge and depth-discharge relations were studied 

in a large sand-bed recirculating flume. From this study, it was found that 

the form of these relations are intimately related to:

1. Regime of flow

2. Form of bed roughness

a. Characteristics of the bed material 

b. Concentration of fine sediment 

c. Temperature

3. Rate of change of discharge with time

In the range of shear where ripples and dunes develop on the bed, the stagedischarge curve for a rising stage is usually quite different from that for a 

falling stage. These curves are valid only for the conditions upon which they 

are based; no general solution is possible. In the range of shear which develops plane bed, standing sand and standing water waves that are in phase, 

and antidunes the rising- and falling-stage curves coincide and hold for all 

values of discharge associated with these forms of bed roughness.

When a shear stress in a channel forms dunes at small discharges and plane 

bed and perhaps standing waves and antidunes at larger discharges, there is a 

discontinuity in the stage-discharge or depth-discharge curves, particularly on 

the rising stage that occurs when the dunes wash out This is caused by the 

great reduction in resistance to flow which occurs when the bed form changes 

from ripples or dunes to plane bad, standing waves, or antidunes and the 

resultant reduction in depth, even though discharge is increasing.

FORMS OF BED ROUGHNESS IN ALLUVIAL CHANNELS

In a rigid channel a well-defined depth-discharge or stage-discharge curve can 

be developed that shows only minor scatter and generally no discontinuities. 

When plotting stage-discharge curves for alluvial streams, this ideal condition 

is not usual even when scour or fill do not occur. Because of the extreme variation in the resistance to flow as discharge and form of bed roughness vary and 

because of the effect of rate of change of discharge with time on the rate of 

development of the various forms of bed roughness, the stage-discharge relation 

for alluvial channels is not well defined. In fact, several different curves are 

possible at a single station unless the shear stress on the bed is always large 

enough that ripple and dunes do not form. If the possible changes in the form

E-l
E-2 STUDIES OF FLOW IN ALLUVIAL CHANNELS

of bed roughness, which cause the shape of the stage-discharge curve to vary 

from time to time and flood to flood, are not known, it is impossible to explain 

the apparent haphazard scatter of points that results.

Two regimes of flow, the lower regime and the upper regime occur 

in alluvial channels. In the lower-flow regime the forms of bed 

roughness that can form are: plane bed (no sediment movement), 

ripples, dunes with ripples superposed, and dunes. In the upper-flow 

regime the forms of bed roughness that can form are: plane bed, 

standing sand waves, antidunes, and violent antidunes. The two 

regimes of flow are connected by a transition zone in which the dunes 

gradually vanish with increasing boundary shear.

These forms of bed roughness were observed in equilibrium experiments, in which a recirculating flume, 8 feet wide and 150 feet long, 

was used. Flume discharge could be varied from 2 to 22 cfs and 

slope from 0 to 1.5 percent. The term "equilibrium" is used to denote 

the conditions that exist after a run has been continued until the 

average slope of the water surface is parallel with the average slope 

of the bed, and the bed configuration is fully established for that 

particular discharge and slope.

In this study the plane bed with sediment moving in the upperflow regime only occurred when the medium size of bed material was 

approximately 0.4 mm and smaller. Standing waves only occurred 

when the median size of bed material, d, was larger than approximately 0.4 mm. The major forms of bed roughness are illustrated 

in figure 1. The major variables that influence the form, of bed 

roughness as discussed by Simons and Richardson (1960) are indicated in equations 1 and 2.

Bed roughness=<fn (D, S, d, a, p, g, a, Sf,fs) (1)

in which

co or dl=fa(d, ps, p, g, Sp, n) (2)

and

D is the depth

S is the slope of energy gradient

d is the median diameter of bed material

d1 is the effective median fall diameter of the bed material

<r is a measure of the size distribution

p is the mass density of the water

g denotes the acceleration of gravity

(a is the fall velocity of the bed material

Sf is the shape factor of the cross section

/, describes the seepage force

p, is the mass density of the bed material

Sp is the shape factor of the particle

p is the viscosity of the water sediment mixture
9

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E-4 STUDIES OF FLOW IN ALLUVIAL CHANNELS

In alluvial channels there is usually either inflow to or outflow from 

the channel through the bank and bed material that causes seepage 

forces. If there is inflow, the seepage force acts to reduce the effective 

size of the sand; and consequently, the stability of the bed material. 

If there is outflow from a channel, the seepage force acts in the direction of gravity and increases the effective size of the sand and 

stability of the bed material. In effect, the seepage force reduces or 

increases the effective size of the bed material by changing its effective 

weight. As a direct result, the seepage forces can influence the form 

of bed roughness and the resistance to flow for a given channel slope, 

channel shape, bed material, and discharge. For example, a bed material with median diameter of 0.5 mm will be molded into the following forms as shear is increased: ripples, dunes, transition, standing 

sand and water waves, and antidunes. If this same material was subjected to a seepage force that reduced its effective weight to a value 

consistent with that of fine sand, which has a median diameter, 

e?=0.3 mm, the forms of bed roughness would be ripples, dunes, transition, plane bed, and antidunes for the same flow conditions. The 

reason for this difference in the forms of bed roughness is the reduction 

in effective weight of the bed material. With the 0.5 mm sand, a 

break in the rating curve normally occurs as bed roughness changes 

from dunes to standing waves. However, with the seepage force in 

effect, a break in the rating curve can occur when the dune condition 

changes to plane bed.

With the seepage force acting with gravity, the 0.5 mm material 

acts as if it were coarser and again there is some effect on form of bed 

roughness and the stage-discharge relation.

A rather common field condition involves outflow from the channel 

during the rising stage, which builds up bank storage and increases 

the stability of the bed and bank material. Then on the falling stage, 

the situation reverses. There is inflow to the channel because of the 

presence of bank storage, which reduces the effective weight and 

stability of the bed and bank material. This field situation can also 

influence the form of bed roughness and the stage-discharge relations.

Simons, Richardson and Haushild have shown (1960, written communication) that the presence of fine sediment in the water, such as 

clay, influences the resistance to flow. Using a bentonite clay, it has 

been determined in the laboratory that with concentrations of this 

type of fine sediment on the order of 40,000 ppm, resistance to flow 

in the dune range is reduced as much as 40 percent. The fine material 

may reduce resistance to flow to an even greater extent under field 

conditions by decreasing the bank roughness as well as the bed roughness, and as previously discussed, any factor that influences bed rough-
BED ROUGHNESS AND DEPTH-DISCHARGE RELATIONS E-5

ness and resistance to flow will likewise alter the form of the stagedischarge relation. Also, fine sediment in the flow may change a 

standing-wave condition into a breaking antidune, which will increase 

the resistance to flow. Thus, the stage-discharge relationships for a 

stream may be different for clear water than for flow heavily laden 

with sediment.

The principal reason for changes in resistance to flow is that the 

viscosity and specific weight of the water-sediment dispersion increases as the concentration of fine sediment increases. This reduces 

the fall velocity of the bed material and its effective size. Form of 

bed roughness and resistance to flow are related to fall velocity; with 

a plane bed, however, physical size also is important.

Changes in temperature can alter the form of bed roughness; and 

hence, the resistance to flow. Vanoni and Brooks (1957) report that 

with higher temperature, there is a decrease in resistance to flow; 

whereas, Hubbell (1956) reports that with higher temperature, there 

is an increase in the resistance to flow. These two apparently contradicting statements, which show the effect of temperature variation, 

are easily explained by considering the forms of bed roughness that 

existed in the alluvial channels in question, and the effect of temperature change on the fall velocity of the bed material. Lowering 

the temperature increases the viscosity of the water and decreases the 

fall velocity of the sand. Thus, a lowering of the temperature decreases the fall velocity of a given sand in the same manner that an 

increase in concentration of fine suspended sediment decreases its fall 

velocity. Consequently, if a sand bed is covered with small ripples 

and the temperature of the water is lowered, the mobility of the particles is increased because of the decrease in fall velocity of the sand, 

larger ripples form, and resistance to flow increases. On the other 

hand, if the form of bed roughness is near to or in transition and there 

is a reduction in the temperature of the water, the fall velocity of the 

bed material is reduced, the given shear causes the dunes to wash out 

to a greater extent or perhaps even causes the bed to become plane, 

which (in either event) is accompanied by a decrease in resistance 

to flow. Both of these phenomena are reversible.

RESISTANCE TO FLOW IN ALLUVIAL CHANNELS

The resistance to flow in alluvial channels, whether in equilibrium 

or otherwise, is largely dependent on the form of the bed roughness. 

This concept has been clearly stated by Leopold and Maddock (1953) 

and Eubey (1952). In general, the resistance to flow is relatively 

small with a plane bed prior to the beginning of transport of bed

621613 62 2
E-6 STUDIES OF FLOW IN ALLUVIAL CHANNELS

material and increases in magnitude with increasing shear stress, 

reaching a maximum value with ripples or dunes, depending on the 

characteristics of the bed material. Eesistance to flow is relatively 

large throughout the range of fully developed ripples and dunes. 

With further increase in shear, the transition zone is reached. Within 

this zone, resistance to flow reduces rapidly with further increase in 

velocity to as little as one-third of its maximum value. The resistance to flow is minimum for a given bed material throughout 

the plane bed and (or) standing wave range. Then as antidunes 

form, the resistance to flow increases with further increase in boundary shear.

A more quantitative concept of the relationship between form of 

bed roughness, resistance to flow, and size of bed material can be 

obtained by referring to figure 2. Note that with the finer sand, the 

maximum resistance to flow occurs hi the ripple range. This condition, which is related to the effect of size of bed material on forms 

of ripples and dunes, is discussed in a paper on forms of bed roughness by Simons and Richardson (1960,1961).

On the basis of a preceding concept of the regimes of flow, the forms 

of bed roughness and resistance to flow in alluvial channels, a realistic 

interpretation of flume depth-discharge curves, and the peculiar variations that occur in them can be presented. These interpretations are 

also generally valid for stage-discharge relations in the field although 

the effect of scour and fill is not accounted for.

In the field, the conditions are such that the variation of discharge 

with time is large, but the corresponding variation of slope is relatively small. Conversely, for laboratory conditions the variation in 

discharge is small (limited by the capacity of the pumping plant) and 

slope can be varied over a large range at will.

EFFECT OF FORM OF BED ROUGHNESS ON DEPTHDISCHARGE RELATIONS

To illustrate the effect of regime of flow and form of bed roughness 

on depth-discharge relations, a series of runs simulating runoff events 

hi which discharge and slope were varied were routed through a recirculating flume with a sand bed. The resultant depth-discharge 

curves resemble the field depth and stage-discharge relations reported 

by Leopold and Maddock (1953), Colby (1960), and Bawdy (1961).

In each run the discharge rate was changed in steps of 3-4 cfs every 

30-45 minutes from a minimum of 5 cfs to a maximum of 21.5 cfs as 

indicated in figure 3. The duration of each run was approximately 8 

hours. For each discharge on both the rise and the recession, the 

stage, depth, velocity, slope of water surface, total-sediment load, and
U 9NINNVW
E-8 STUDIES OF FLOW IN ALLUVIAL CHANNELS

water temperature were continuously measured. The form of bed 

roughness and its variation with discharge and time was carefully 

observed, described, and photographed. The characteristics of the 

bed material used in each run were determined beforehand. The 

methods used to measure the variables are described in detail by 

Simons, Richardson, and Albertson (1961). To observe the variation 

of the measured variables with time as discharge was changed, refer 

to figure 3. The accuracy of the discharge measurements were excellent. The accuracy with which the other variables such as slope, 

total load, velocity, stage, and depth were measured was limited because of the nonequilibrium condition of the flow, the relatively small 

number of measurements upon which the magnitude of a variable was 

based, and the natural variation of these variables with time and distance. For example, it has been verified by the writers that it is 

necessary to sample total-sediment load almost continuously over a 1- 

to 2-hour period in order to determine accurately total-sediment transportation. In the nonequilibrium runs, sampling could only be done 

over a 15- to 30-minute interval; and in addition, sediment load varied 

more radically with time than normally because once the run was 

started, there was never time for the channel to become completely 

stable before the discharge was changed.

In spite of the limited accuracy with which some of the variables 

were measured, data for depth, discharge, and form of bed roughness 

can be utilized to illustrate the importance of form of bed roughness 

on rating curves for depth and discharge. Depth-discharge relationships were used because they eliminate changes that would occur in 

stage-discharge relationships resulting from scour and fill. The importance of form of bed roughness on the depth-discharge relations is obviously equally valid for stage-discharge relations. Einstein 

and Chein (1958) predicted some of the forms of depth-discharge 

curves presented in the following sections. However, contrary to 

present evidence, they did not believe they were of practical importance.

LOWER-FLOW-REGIME RtTN WITH CONSTANT CHANNEL SLOPE

The run upon which figure 4 is based was in the lower-flow regime. 

The sand bed had a median fall diameter of 0.45 mm. Flume conditions were such that the slope was held nearly constant throughout the 

duration of the run. The initial discharge of 5.04 cfs was set up 12 

hours prior to the beginning of the run in order to start with stable 

channel conditions. The initial form of bed roughness, at minimum 

discharge, was small dunes with ripples superposed. As the discharge 

was increased, the ripples were gradually eliminated and larger and
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DISCHARGE, 0, IN CUBIC FEET PER. SECOND

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BED ROUGHNESS AND DEPTH-DISCHARGE RELATIONS E-ll

larger dunes were developed by the flow. At the peak discharge of 

20.96 cfs, the form of bed roughness was large dunes. After reaching 

maximum discharge, the flow was reduced in steps to produce a nearly 

symmetrical variation of discharge with time as illustrated in figure 3. 

The depth-discharge curve for increasing discharge is lower than the 

depth-discharge curve for decreasing discharge. This situation can 

be explained in terms of the bed roughness and its changes with discharge and time. With increasing flow, the development of maximum 

roughness lagged behind the increase in discharge. That is, the type 

of bed configuration consistent with a particular discharge did not 

have time to develop fully before discharge was increased again. The 

net result was that the resistance to flow lagged behind the increase 

in discharge, and depth was less for a given discharge than it would 

have been if equilibrium had been established.

The opposite condition prevails for this run during recession. 

That is, when discharge was decreased, the flow could not fully alter 

the large dunes to smaller ones required for equilibrium in the limited 

time available. The net result was that the channel was rougher 

and deeper than it should have been for a given discharge, and the 

recession curve of the depth-discharge relation was above the rising 

curve of the same relation. The depth at the conclusion of the run 

was larger than the initial depth even though the discharge was the 

same. However, continuing the run for a few hours with the final 

discharge, the depth did return to its original value.

The magnitude of the spread between the two branches of the depthdischarge relations is dependent primarily on variation of roughness, 

which is influenced by rate of change of discharge with time, the 

characteristics of the bed material, and to a limited extent on concentration and type of fine sediment and other variables that are discussed later. If the rate of change of discharge with time is large, 

the spread between the two branches of the depth-discharge curve 

may be large or relatively small. If the rate of change of discharge 

with time is very small, the spread between the two branches will be 

small. At some intermediate rate of change of discharge with respect 

to time, the spread between the two curves will be maximum.

The size of the bed material is an important variable. With fine 

sand, the rate of change of bed configuration with time is faster than 

for a coarse sand, and the change from a dune bed to a plane-bed 

configuration occurs at smaller shear because of the increased mobility of the bed material. Hence, the spread between the rising and 

falling curves of a depth-discharge relation for a fine sand can be 

larger or smaller, and a break in the rating curve develops at a smaller 

depth for a particular runoff event than that for a coarse sand.
E-12 STUDIES OF FLOW IN ALLUVIAL CHANNELS

LOWER- AND UPPER-FLOW-REGIME RUNS 

SMALL CHANNEL, SLOPE VARYING WITH DISCHARGE

Another depth-discharge relation for the type of runoff event illustrated in figure 3 is presented in figure 5. There was no tailgate control at the end of the flume, the slope-of-energy gradient varied 

directly with discharge, and the median diameter of bed material was 

0.28 mm. As in figure 4 for increasing discharge, the development 

of resistance to flow lagged behind the increase in discharge until 

Q ^ 19 cf s. A common curve described the depth-discharge relation 

for both increasing and decreasing discharge when Q ^ 19 cfs, and 

the recession curve was above the rising depth-discharge curve when 

Q < 19 cfs. This latter condition indicates an excessive resistance 

to flow caused by residual-bed roughness. That such was probably 

the case, can be verified by considering variation of form of bed roughness with discharge. As indicated in figure 5 at minimum discharge, 

the initial bed roughnes was ripples. As discharge was increased, the 

bed form changed to ripples superposed on dunes, then to large dunes 

and finally the transition zone developed that was followed by a plane 

bed. With decreasing discharge, the bed roughness was first modified 

from the plane bed to transition condition and then back to large 

dunes. With further reduction in discharge, the resistance to flow 

was greater than normal because of the residual dunes. At minimum 

discharge, the bed roughness was ripples superposed on small dunes 

and residual dunes. The minimum flow was maintained for a period 

of 12 hours beyond completion of the 8-hour run. At the end of this 

time, the residual dunes had disappeared and the rising and falling 

depth-discharge curves coincided giving the closed-loop relationship 

presented in figure 5. It is important to note near the maximum 

discharge, that after the change from lower to upper flow regime 

the depth-discharge curve was the same for both increasing and decreasing discharges.

MODERATE CHANNEL SLOPE VARYING WITH DISCHARGE

The physical conditions associated with this upper- and lower-flowregime run with slope varying with discharge were the same as those 

described for figure 5 except that channel slope was steeper. Referring to figure 6, the initial bed roughness at the beginning of the 

run (stable flow) was dunes with ripples superposed. As the discharge was increased, ripples were eliminated and larger dunes developed. With further increase in discharge the form of bed roughness gradually changed from dunes to a plane bed. The curve connecting the points for the plane-bed condition was displaced to the
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133J Ml Y7 'Hid3a
E-14 STUDIES OF FLOW IN ALLUVIAL CHANNELS

right appreciably from the increasing discharge curve for ripples 

superposed on dunes and dunes. That is, the change in resistance to 

flow, which occurred as the form of bed roughness, changed from 

dunes to plane bed caused a significant break in the rating curve. 

This break in the depth-discharge curve is typical of the breaks in the 

stage-discharge curves of most alluvial channels in which the form of 

bed roughness changes from ripples and (or) dunes to plane bed, 

standing waves, or antidunes (Colby, 1960, and Dawdy, 1961).

The curve representing the decreasing depth and discharge condition shows that there is a common curve for increasing and decreasing 

discharge with upper-regime flow. However, the change from dunes 

to plane bed takes place at a larger discharge as discharge is increasing than the change from plane bed to dunes when discharge is decreasing. In fact, in this run it was difficult for the bed to change 

from a plane bed to dunes in the time allowed between changes 

in discharge. That is, dune and ripples never developed with decreasing discharge to the extent that they did with increasing discharge. Hence, resistance to flow and depths were smaller for 

small discharges as depth was decreasing than they were for the 

corresponding discharge when depth was increasing with time. 

Note that the reverse condition was true in figure 4 and figure 5. 

However, the foregoing condition could be reversed by using a 

longer time period between changes in discharge that is, if rate 

of change of discharge with respect to time was decreased. Under 

this condition, dunes would have time to develop and the increasingand decreasing-depth curves in the ripple and dune range of bed 

roughness would be closer together. In fact, if the rate of change of 

discharge was much slower with decreasing depth than with increasing depth, the curve that represents decreasing discharge and 

depth would probably cross and be above the increasing-discharge 

curve. Thus, rate of change of discharge with respect to time is 

extremely important in defining shape of the depth-discharge relation. 

At minimum discharge, at the end of the run, the point on the decreasing-depth curve was considerably below the corresponding point 

for equilibrium conditions on the increasing-depth curve. Holding 

discharge constant, this point moved vertically upward with time 

(depth increased) until, when equilibrium conditions were again 

reached, the two points coincided.

LARGE CHANNEL SLOPE VARYING WITH DISCHARGE

The conditions for the run shown in figure 7 were similar to those 

that yielded figure 6, except that the size of the bed material was 

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E-16 STUDIES OF FLOW IN ALLUVIAL CHANNELS

and the variation of discharge with time was almost symmetrical as 

illustrated in figure 3. In figure 7 at the beginning of the run, the 

bed configuration was regular dunes of medium amplitude. As the 

discharge was increased, large dunes, transition, and finally standing 

waves developed. As the form of bed roughness changed from dunes 

to standing waves, there was a significant break in the depth-discharge curve. To the right of the break in the upper flow regime, the 

increasing and decreasing discharge relation can be represented by 

a single curve. This part is similar to the depth-discharge relation 

illustrated in figure 9, and the part of figure 6 to the right of the 

loop. With decreasing discharge in the vicinity of and immediately 

to the left of the break in the rating curve, dunes were reforming; 

but the size of the dunes and, hence, the magnitude of resistance to 

flow was less than for comparable discharge with increasing depth. 

With further reduction of discharge, dunes developed to the point 

where at a still smaller discharge, a residual-dune effect was apparent, 

that is, dunes remained that were larger than they were for comparable discharge on the increasing depth curve; hence, the decreasing 

depth-discharge relation crosses the increasing depth-discharge relation and remains above it because of the extra resistance contributed 

by the residual dunes. At the termination of the run, the point on 

the decreasing depth curve continued to drop with time until it 

matched the initial point on the increasing depth curve completing 

the loop as shown. The return to the original depth with continued 

minimum flow, after completion of the runs (figs. 5,6, and 7), indicates 

the accuracy of the data and the ability to accurately repeat an 

equilibrium run.

This depth-discharge curve also indicates that size of bed material 

and rates of change of discharge with time are closely related to the 

shape of the depth-discharge relation in the ripple and dune range of 

bed roughness. Thus, an infinite number of possibilities exist in the 

ripple-dune range. If the rate of change of discharge with time 

was faster, the decreasing discharge curve might not cross the increasing discharge curve at all; or if the rate of change of discharge 

was very slow, the decreasing discharge curve would cross the increasing discharge curve at a larger discharge. Similarly, if the rate of 

change of discharge with increasing depth was different, the position 

of this curve would be changed; and for a different size of bed material, the break in the rating curve would occur at a different 

discharge.

E-18 STUDIES OF FLOW IN ALLUVIAL CHANNELS

UPPER- AND LOWER-FLOW-REGIME BUN WITH CHANNEL SLOPE 

VARYING INVERSELY WITH DISCHARGE

To further illustrate the numerous types of depth- and stage-discharge relations that are possible in alluvial channels, consider the 

situation where there was a channel constriction at the end of the 

flume. The channel slope and size of constriction were of such a nature 

that at small discharges flow was rapid; and as discharge was increased, the constriction caused tranquil-flow conditions to develop. 

This condition could exist in the field upstream from a bridge, culvert, 

or natural constriction. The depth-discharge relation for this condition is illustrated in figure 8. At the beginning of the run, flow was 

rapid. There were small in-phase standing sand and standing water 

waves as well as antidunes. As discharge was increased, the magnitude of the Froude number soon began to decrease because of the backwater effect of the constriction and dunes began to form. With still 

further increase in discharge, the dunes increased in size and the largest dunes occured at maximum discharge. As magnitude of discharge 

decreased, large residual dunes persisted, causing a greater resistance 

to flow with decreasing discharge than that which existed at the same 

discharge with increasing depth. With further reduction in discharge, 

the effect of the backwater curve lessened and the Froude number increased until the transition condition (washed-out dunes) developed. 

With still further reduction in discharge, upper-flow regime conditions developed with standing sand and standing water waves and 

some antidune activity.

The depth-discharge relation of figure 8 is practically the reverse of 

the ones illustrated in figures 5 and 6.

As with the other depth-discharge relations presented herein, the 

form of the curve or curves is intimately related to the rate of change 

of discharge with time and the size and gradation of the bed material. 

Except for plane bed, standing wave, and antidune conditions (upperflow regime) the depth-discharge relations can only be identical for 

runs and physical conditions that are identical.

UPPER-FLOW-REGIME RUN

The most simple form of depth-discharge curve in alluvial channels 

exists when only plane bed, standing waves, and antidunes develop 

(no ripples, dunes or transition). This condition was investigated 

in the laboratory using a bed of sand of a median diameter of 0.45 

mm. The resultant depth-discharge curve is presented in figure 9. 

At minimum flow on the increasing discharge curve, the bed roughness 

consisted of plane bed, standing waves, and very limited antidune 

activity. As discharge was increased, the antidune activity increased
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DISCHARGE, 0, IN CUBIC FEET PER SECOND

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DISCHARGE, 0, IN CUBIC FEET PER SECOND 

FIGURE 9. Relation of depth to discharge for upper-regime flow.
BED ROUGHNESS AND DEPTH-DISCHARGE RELATIONS E-21

until at peak discharge, there was very strong antidune activity. With 

decreasing discharge, the process reversed itself. That is, antidune 

activity diminished to the original plane-bed, standing-wave, and mild 

antidune condition. There is no break in this rating curve. The 

depth-discharge data plot on a common curve, excluding error in 

measurement of depth. This indicates that depth-discharge curves for 

field conditions should be reasonably accurate when the plane bed, 

standing waves, and antidunes are the only forms of bed roughness. 

The accuracy of this type of depth-discharge relationship is also 

illustrated in figure 7 in that part of the relation to the right of the 

break. With finer sand (d^Q:3 mm) there was a relatively rapid 

increase in resistance to flow with increasing Froude number for 

antidunes. This may increase the scatter in depth-discharge relations 

for flumes and streams with beds of fine sand.

OTHER FACTORS THAT INFLUENCE DEPTH- AND 

STAGE-DISCHARGE RELATIONS

Thus far, the importance of the physical conditions existing in a 

channel including the rate of change of discharge with time and the 

characteristics of the bed material on depth-discharge and stagedischarge relations have been emphasized. There are other factors 

that can be of at least minor importance, such as, bank vegetation, 

scour and (or) fill, the bulking or consolidation of the sand bed as 

changes occur in the form of bed roughness, multiple roughness, 

and wind action. These additional factors will be discussed qualitatively.

Bank vegetation which trails in the water cross section and (or) 

aquatic plants can appreciably increase resistance to flow in natural 

channels. Thus the presence of vegetation in an alluvial channel 

can delay or even eliminate the break in a rating curve. Vegetation 

can also change the spread between rising-stage and falling-stage 

curves within the lower-flow regime. For example, the channel vegetation presents a large resistance to flow on the rising stage, increasing 

depth until the vegetation is plastered down by the action of the 

water-sediment complex. Then on the falling stage, after the plastering down and the possible removal of some vegetation, the resistance to flow is relatively smaller. Considering the superpositioning 

of a vegetation effect on the depth-discharge relation of figure 5, it 

is conceivable that the spread between the two curves would be reduced or at least slightly changed, depending on the characteristics 

and behavior of the vegetation.

Under certain physical conditions, the reach for which a stagedischarge relation is being developed may tend to scour or aggrade
E-22 STUDIES OF FLOW IN ALLUVIAL CHANNELS

during a flood. This change in bed elevation naturally effects the 

stage-discharge relationship. Also, the resultant change in bed elevation can influence bed roughness and the depth-discharge relation for 

the reach. Consider, for example, figure 8. The stream has a relatively large capacity to transport sediment per unit of flow at small 

discharges. As discharge increases, the backwater effect develops and 

reduces the transport capacity in the backwater area. A large sediment load is thus carried into a reach of reduced slope and velocity, 

and as a result, the bed of the channel begins to aggrade seeking a new 

equilibrium level. With increasing depth, the bed level increases 

with time, and on the falling stage, it continues to rise with time at a 

diminishing rate until the backwater effect is eliminated. Then, because of the aggraded bed, a steeper than normal channel slope can 

exist in the reach causing a larger Froude number and increased 

transport of sediment in the aggraded reach until the original equilibrium is restored.

The bed material of an alluvial channel is fairly loose and soft 

when ripples exists. It becomes even softer and bulks to an even 

greater extent as dunes form. When the transition zone is reached 

(washed-out dunes), the bed becomes firmer and its weight per unit 

volume increases. With further increase in shear stress the bed continues to become firmer until with plane bed and (or) standing sand 

and standing water waves, the bed has a maximum density. With 

still further increase in shear, antidunes develop. As larger and more 

violent antidune action occurs, the bed density may decrease slightly. 

This variation in density with bed congfiuration causes minor 

changes in bed elevation which can, at least slightly, affect stage-discharge relations. This concept is applicable to all breaking stagedischarge relations but can have only a very minor effect when limited 

to ripple and dune type configurations in the lower-flow regime. By 

working with depth instead of stage, this effect is automatically eliminated. Generally it is quite difficult to measure and record depth, 

but this problem may be eliminated eventually by using a sonic depth 

sounder to record bed elevations. A sonic depth sounder developed 

by Eichardson, Simons, and Posakony (1961) does an excellent job 

of recording bed elevation under laboratory conditions and it should 

be possible to redesign this instrument so that it will measure and 

record depth of flow in the field.

The forms of bed roughness that were observed on the bed of the 

laboratory flume were consistent across the full width of the flume. In 

the field this may not be true. A multiple roughness pattern may 

exist. At the middle of the stream, the bed may be plane or have
BED ROUGHNESS AND DEPTH-DISCHARGE RELATIONS E-23

antidunes; whereas at the sides, there may be dunes. With these 

multiple roughnesses, the bed would be firm in the middle and soft 

at the sides of the channel. Another possible combination of roughness elements is antidunes and (or) plane bed on one side of the 

stream with dunes on the other and this situation may reverse with 

time at the same site. As changes occur in the magnitude and areal 

extent of the various forms of bed roughness in a stream, the depthdischarge, and consequently, the stage-discharge relation will also 

change during a runoff event.

Depending on the way in which the wind is oriented with respect to 

the reach of the channel in question, the resistance to flow may be 

increased or decreased, and in every instance channel stability is 

decreased. Wind in the opposite direction to flow can increase resistance to flow an appreciable amount. Wind in the direction of flow 

slightly decreases resistance to flow. Wind in either direction causes 

instability of banks because of the wave action it generates and reduces the stability of bed material by causing seepage forces. The 

problem of flow in a homogeneous alluvial bed, as a result of wave 

action above the bed has been analyzed by Putnam (1949). Using 

Putnam's results Simons has qualitatively verified that water surface 

waves generated by wind can reduce the effective weight of bed material by as much as 20 percent as a result of the additional tractive 

force exerted on the bed material because of the water waves and 

seepage forces set up in the bed because of difference in elevation between wave crests and troughs.

Because the wind action can affect the stability of the bed material 

and change the resistance to flow, it probably can also affect the 

stage-discharge relations.

The influence of changes in bed configuration, resistance to flow, 

and rate of change of discharge on depth-discharge relations for 

alluvial channels has been emphasized. All loops in the curves discussed were obviously the result of changes in bed roughness. However, there are also loop-rating curves, under certain circumstances, in 

rigid boundary streams. The magnitude of the spread between the 

two curves forming the loop depends on the slope of water surface, 

magnitude and velocity of the flood wave, rate of change of discharge 

with time, channel storage, and overbank storage (Corbett and others, 

1945). These factors will also affect the stage-discharge relationship 

for alluvial channels in the same manner as in rigid channels, but in 

addition, they may play a more complex role since they may also 

change the bed roughness and the resistance to flow.
E-24 STUDIES OF FLOW IN ALLUVIAL CHANNELS

CONCLUSIONS

The resistance to flow in an alluvial channel under all conditions of 

flow is intimately related to the form of bed roughness. In turn, the 

form of bed roughness varies with such factors as the magnitude of 

the shear stress exerted by the water on the bed, the characteristics 

of the bed material, the fine sediment load, seepage forces caused by 

flow through the bed and bank material, and temperature.

As the form of bed roughness changes, the magnitude of the resistance to flow can change as much as 300 percent. For example, considering flume conditions and a plane or standing wave bed configuration the Manning^ can be as smallas Q-PJjj but by slightly decreasing 

the shear stress the~pTalien5e^olF^tandlng^ waves vanish and a dunebed condition results that can have a MaiinJ.n^,«cJ^Jaige^sjO.,036.

The large changes in resistance to flow, which occur asaresult of 

changing the form of bed roughness, influences the form of the stagedischarge or depth-discharge curves for alluvial channels.

There are, in general, three types of stage- or depth-discharge relationships for alluvial channels. lT^n^lb^3~ToTrm~1i3ways consists "of 

some combination of ripples, dunes, or washed-out dunes regardless 

of variation in discharge, then consecutive measurements of depth and 

discharge made during one flood, when plotted, may loop or even cross 

forming two loops. That is, the depth for a given discharge on the 

increasing discharge curve will usually be larger or smaller than the 

depth at the corresponding discharge with decreasing discharge conditions. These loop-rating curves for increasing and decreasing discharge conditions are infinite in number for a given reach of channel. 

Each new runoff event yields a new set of curves unless the flows and 

channel conditions are identical. Hence, stable depth-discharge relationships do not exist under these conditions.

The reason for the loop or multiple loop depth-discharge curves is 

that the formation and alteration of roughness elements for each 

change in shear stress, with the increase or decrease in depth, will lag 

behind the change in depth. For instance, with increasing discharge, 

the dunes do not increase in size at a rate corresponding to the rate at 

which the discharge and shear stress are increasing. Consequently, 

the resistance to flow will be less than under equilibrium flow conditions. Whereas, with decreasing discharge the rate of decrease of the 

large roughness elements formed at the greater depth (residual roughness is lagging) lags behind the decrease in discharge and the resistance to flow is greater than it would be for equilibrium conditions. 

Thus, the curve for increasing depth is below the curve for decreasing 

depth in this instance.
BED ROUGHNESS AND DEPTH-DISCHARGE RELATIONS E-25

The reverse situation, where the increasing-discharge curve is above 

the decreasing-discharge curve, can also be explained in a similar 

manner by the lag in the change of resistance to flow with respect to 

change in discharge. However, in this situation the resistance to flow 

is greater with increasing depth than with decreasing depth.

The shape of the loop curves depends on the form of bed roughness that forms under equilibrium conditions for a given depth, the 

amount of time it takes for the bed roughness consistent with equilibrium conditions to form, and the amount of time available for the bed 

to adjust before the depth changes as a result of additional change in 

discharge. The above three factors are determined primarily by the 

energy gradient, fall velocity of the bed material, and the rate of 

change of discharge with time.

The secondtype of depth-discharge curve occurs when the bed 

formisli3ways~plane, standing waves, or antidunes regardless of the 

discharge. In this type of curve, the slope of the energy grade line 

is steep enough and the shear stress is great enough, for the effective 

size of the bed material involved, so that only these forms of bed 

roughness can develop. For these conditions, the depth-discharge 

curve is the same for both increasing and decreasing discharge and is 

stable and reliable.

The thirdgeneral relationship results when the shear varies over 

such a wicle rangeHEhat ripples and (or) dunes form at small discharges, and plane bed, standing waves, or antidunes form at large 

discharges. With these conditions, the lower part of the depthdischarge relationship will be the same as the first type of general 

curve including all of its variability. Then, there will be a break 

in the relationship at some intermediate discharge, which leads to the 

second type of curve. This break will be downward and to the right 

on a plot of depth versus discharge with increasing depth, because 

of the decrease in resistance to flow as the bed roughness changes, and 

it will be upward and to the left on the recession curve because of the 

increase in resistance to flow. In accordance with the description of 

the type two curve, there is little variation in the depth-discharge relationship to the right of the break. However, the break in the depthdischarge curve will not occur at the same discharge on the recession 

as it did on the rise, or at the same discharge from one runoff event 

to another. Where the break occurs in the relationship depends on 

the discharge, rate of change of discharge with time, and the characteristics of the bed material; its position may vary from rise to rise.

Stage-discharge relationships are influenced by all the variations 

that occur in depth-discharge relationships in addition to those resulting from local scour and fill as well as bulking or consolidation
E-26 STUDIES OF FLOW IN ALLUVIAL CHANNELS

of the bed material. Thus stage-discharge curves can be similar to 

the depth-discharge curves presented or quite different depending on 

the amount of scour and fill; the porosity of the bed, which is related to form of bed roughness; and the time element involved.

The form of the stage or depth-discharge curves can be changed 

slightly, particularly the position of the point of discontinuity by the 

presence of seepage forces, large concentrations of fine sediment in 

the stream, wind action, vegetation, temperature change, rate of change 

of stage with time, multiple roughness, and unsteady flow.

With an understanding of the basic principles that cause the observed variation in depth-discharge and stage-discharge relations, a 

more effective use of these relationships is possible.

LITERATURE CITED

Colby, B. B., 1960, Discontinuous rating curves for Pigeon Roost and Ouffawa

Creeks in northern Mississippi: U.S. Dept Agr. ARS 41-36. 

Oorbett, D. M., and others, 1945, Stream-gaging procedure: U.S. Geol. Survey

Water-Supply Paper 888. 

Dawdy, D. W., 1961, Depth-discharge relation of alluvial streams; discontinuous

rating curves. Water-supply Paper 1498-C. 

Einstein, H. A., and Chein, N., 1958, Discussion of mechanics of streams with

moveable beds of fine sand: Am. Soc. Civil Engineers Trans., v. 123, p.

553-562.

Hubbell, D. W., and others, 1956, Investigations of some sedimentation characteristics of a sand bed stream: U.S. Geol. Survey open-file report, Lincoln,

Nebr. 

Leopold, L. B., and Maddock, T., Jr., 1953, The hydraulic geometry of stream

channels and some physiographic implications: U.S. Geol. Survey Prof.

Paper 252. 

Putnam, J. A., 1949, Loss of wave energy due to percolation in a permeable

sea bed: Am. Geophys- Union Trans., v. 30, no. 3. 

Richardson, E. V., Simons, D. B., and Posakony, G. J., 1961, Sonic depth sounder

laboratory and field use, U.S. Geol. Survey Circ. 450. 

Rubey, W. W., 1952, The geology and mineral resources of Hardin and Brussels

quadrangles, Illinois: U.S. Geol. Survey Prof. Paper 218. 

Simons, D. B., and Richardson, E. V., 1960, Resistance to flow in alluvial

channels: Am. Soc. Civil Engineers Jour., v. 86, no. HY 5.

 1961, Forms of bed roughness in alluvial channels: Am. Soc. Civil Engineers, Proc. v. 87, no. HY 3. 

Simons, D. B., Richardson, E. V., and Albertson, M. L., 1961, Flume studies

using a medium sand (0.45 mm) U.S. Geol. Survey Water-Supply Paper

1498A. 

Vanoni, V. A., and Brooks, N. H., 1957, Laboratory studies of the roughness and

suspended load of alluvial streams: California Inst. Technology, rept. E.

68, Pasadena, Calif.

o