Calculation procedure for sand transport by wind on natural beaches

US. Army Caast.Eng: Reo. Cte
M.P. 2-64

U.S. Army
Coastal Engineering
Research Center

CALCULATION PROCEDURE FOR
SAND TRANSPORT BY WIND
ON NATURAL BEACHES

MISCELLANEOUS PAPER No. 2-64
April 1964

i ee ee
WOODS HOLE
OCEANOGRAPHIC INSTITUTION
MAY 2 8 1964

: WOODS HOLE, MASS.
No) Tl
Ng
no. 2-6Y

ow. DEPARTMENT OF THE ARMY
|| CORPS OF ENGINEERS

FOREWORD

Sand transport by wind is a major factor involving stability
of the beach and backshore in some areas. Experimental work on
this subject which validated findings of previous investigators
with respect to the rate of such transport by wind, was presented
in earlier U. S. Army Corps of Engineers publications (Technical
Memorandum No. 119 of the former Beach Erosion Board - "Sand
Movement by Wind Action: or Characteristics of Sand Traps", and
Technical Memorandum No. 1 of the Coastal Engineering Research
Center - "Sand Movement by Wind"). The brief report herein sum-
marizes available methods for calculating the actual rate of sand
transport by wind and presents specific procedures and calculations
for annual transport from the beach inland by wind at a natural
beach locality in California.

This report was prepared at the Wave Research Laboratory of the
Institute of Engineering Research of the University of California
at Berkeley in pursuance of contract DA-49-055-civ-eng-63-4 with the
Beach Erosion Board which provides in part for the study of sand
movement by wind. The author of this report, Abdel-Latif Kadib,
was a graduate student at the University at the time this work was
accomplished,

This report is published under authority of Public Law 166,
79th Congress, approved 31 July 1945, as modified by Public Law 88,
172, approved 7 November 1963.

TABLE OF CONTENTS

INTRODUCTION
SAND TRANSPORT BY WIND .
WIND VELOCITY ABOVE A SAND SURFACE
APPLICATION TO NATURAL BEACHES
Transport Calculations ,
Wind Duration per Year "t" .
Length of Reach Contributing to inland Gieangnae’:
Shear Velocity U,
Total Annual Transport

REFERENCES

TABLES

Page

CALCULATION PROCEDURE FOR SAND TRANSPORT
BY WIND ON NATURAL BEACHES

by

Abdei-Latif Kadib
University of California

INTRODUCTION

The estimation of the annual amount of sand transported along the
coast is important for planning and constructing coastal structures.
One of the motive forces for transporting sand along the coast is the
well-known littoral current generated by wave action, and the other is
the wind. Sand movement by wind action has been treated by several
research workers. In this report, a summary of some of the available
methods of calculating rate of sand transport by wind, and calculations
for annual sand transport inland by wind from natural beaches are pre-
sented.

SAND TRANSPORT BY WIND

Several investigators have developed expressions for the rate of
sand movement as a function of certain variables. Some of these ex-
pressions are as follows:

Ww
Bagnold Formula: ‘2) The rate of sand movement per unit width
and unit time, q, is given by:

Foe AS tu? (1)

where D is the grain diameter of standard 0.25 mm sand, d is the grain
diameter of sand in question, Y is the specific weight of air, U, is the
shear velocity, and c has the following values:

* References on page il.

1.5 for a nearly uniform sand
1.8 for a naturally graded sand
2.8 for sand with a very wide range of grain diameter

(5)

Kawamura Formula: The rate of sand movement, q, is given by:

Q) = ae (Uy — Ux) (U, sts Uae (2)

where y is the specific weight of air, U, is the shear velocity, and U
is the threshold shear velocity, and K is a constant which must be
determined by experiment.

Bait

Gio)

O'Brien and Rindlaub Formula: O'Brien and Rindlaub proposed
the following formula from data derived by field tests

G = 0.036 Ux? (for Us > 20 ft/sec) (3)

where G is the rate of transport in pounds per day per foot width, and

Us is the wind velocity at 5 feet above the sand surface in ft/sec. How-
ever, the use of this formula should be limited to sand having the same
grain diameter of that existing in the field tests‘) (0.195 mm).

WIND VELOCITY ABOVE A SAND SURFACE

The shear stress, T, proauced at the sand surface by wind is one
of the most important factors in investigating sand movement by wind
action. When the shear stress exceeds a certain critical value, the sand
particles start to move. As long as there is no sand movement, the wind-
velocity distribution can be described adequately by the general equation

U =C Log = (4)

in which U is the velocity at height Z above the sand surface and Z) is
a reference parameter. The coefficient, C, according to von Karman's
development, is equal to a U,, where K is the Karman constant, U, is

the shear velocity defined as p and p is the density of air. For

K equals 0.40, the von Karman equation becomes

W = Se7S Wa weg me (5)
Zo

4 :
Concerning the roughness factor, Zo, Zingg ) proposed the equation

2. = O,08 tog (6)
0.18

where Zand the sand grain diameter, d, are expressed in mm. Once the
wind velocity is great enough to move sand particles, the velocity pro-
files for different wind speeds seem to meet at a certain point, which
he called a "focus."" The height of the focus, Z', appears to be
associated with the height of the ripples which form on the surface.
Studies made by Zingg allow one to predict the focus by means of the
formula,

Ze 10d millimeters (7)

Ui 20d miles/hour (8)
where the grain diameter, d, is expressed in millimeters. Thus, using
the component of the focus, Z', U', the wind-velocity distribution can
be expressed by

Ul =1C opee— Ue (9)
Z

Bagnold assumed a coefficient C of 5.75 U,, which corresponds to the
value of 0.40 for the Karman constant. But the experiments by Zingg
yielded the equation

0 = Gite Un to 2a yy (10)
7A

which indicates values of 0.375 for the Karman constant.

APPLICATION TO NATURAL BEACHES

An illustration of the application of the methods of calculating
sand transport by wind was made for Salmon Beach near Bodega Head in
northern California (Figure 1). Sand samples were taken at the mid-tide
level, or reference point, for eight localities along the coast from
Salmon Creek to Mussel Point, a distance of more than 2 miles. Figure 2

BODEGA
HEAD

FIG. | NUMBER AND LOCATION OF REACHES
AND SAND SAMPLES
4

000'0!

SAJIdWVS HOVAS JO SISAIVNVY TIVOIINVHOSAW 2 Sls

SNOYDIN ‘3ZIS NIVY9

Q00S GOOF C002 Ooo 00S 00g 001

e2cesee

eRe as , i
ara’ o/s
PAYA Tt

ag A TE
Ta oe
Ara 1B

Ol 9102 8222 28+ 09 O8oO0!
IdW ‘S3ZIS 3AZIS

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NVHL YSNI4

WAD

° Yo

shows the grain size distribution of sand.* The mean diameter of the
sand (dsq) was found to vary from 1.3 mm at Salmon Creek to 0.35 mm
halfway between Salmon Creek and Mussel Point. According to the grain
size variations and the alignment of the coastline in the area con-
sidered, the whole distance was divided into eight reaches (Figure 1).
Table 1** shows the characteristics of each reach.

Transport Calculations. At this point, one should ask, what method
is to be used for calculating the transport? It is clear from previous
work(2,6) that the Bagnold formula seems to be superior to any other
formula for the following reasons:

1. Bagnold's equation considers the grain-size diameter
(Equation 1),and since we have a significant change
in d5qg from reach to reach, the Bagnold formula seems
superior.

2. The value of the coefficient C in the Bagnold formula
is better defined and more limited in range than the
coefficient K in the Kawamura formula.

3. The Kawamura formula (Equation 2) also includes the
threshold shear velocity which introduces a further
uncertainty in the calculations of transport rate,
especially since the factor is influenced by the
moisture content of the sand.

4. The use of the O'Brien and Rindlaub formula is not
good here, since it has been shown that their equation
should be limited to sand having the same grain
diameter of that tested in the field, (©)(d, = 0.195 mm )

Accordingly, the Bagnold formula will be used in the following calcula-
tions for sand transport.

Equation (1) gives the transport in pounds per second per one-foot
length. Rewriting Equation (1) in a more general way

Oe ee oe (11)

*Since the sand size of the reference point is a measure of the sand
being moved along the coast as littoral drift, it is also the sand
that is moved back into the dune area by wind action.

**Tables at end of text.

where
Q = total transport in pounds per year
C = Bagnold constant

£ = length of reach in feet perpendicular to direction
of wind considered

d = average grain diameter of sand considered (ds5g mm)

~)
u

average grain diameter of standard 0.25 mm sand
Y = specific weight of air = (0.076 1bs/ft?)
U, = shear velocity in ft/sec

T = duration of wind in seconds per year

g = acceleration due to gravity = 32.2 ft/sec?

Now substituting the values of y, g, and choosing C.=1.8, since the sand
considered has a natural grading, we obtain from Equation (11)

Qa 1s oy - s50 100, /2) — On0mee yes
D 32.2
d 3
QF= ss 720) te 5 U, in pounds per year (12)

where t is in hours per year.

Wind Duration per Year "t''. Duration in hours of winds of various
speeds from various directions was collected from data obtained from the
Pacific Marine Station, Dillon Beach, California. Data were available
for the period September 1, 1962 to August 31, 1963 which gave us one
year of records. These data are shown in Figure 3. In Table 2 these
data are summarized for calculation purposes. Wind speeds below 10 mph
were considered calm, since their contribution to transport may be
neglected. The uneven values of wind speeds shown in both Figure 3 and
Table 2 resulted from the reduction of the wind data from the anemometer
chart and the calibration curve of the anemometer. Total number of hours
contributing to transport (greater than 10 mph) was 1135 hours; calm hours
3555 and for 4070 hours the anemometer was inoperative.

24.6 mph (©) 25.7mph

© WIND DURATION
IN HOURS PER
YEAR

26.7 mph 27.7mph (©) 528 mph

FIG.3 DURATION OF WIND IN HOURS PER YEAR FOR DIFFERENT WIND
VELOCITIES AS OBTAINED FROM PACIFIC MARINE STATION

Length of Reach Contributing to Inland Transport.

eight different possible
wind directions, that is,
NSUNEN ES SE. S. “SW. Wi
and NW, it seems that
only four directions
cause inland sand trans-
port at Salmon Beach.
These directions are N,
NW, W, and SW. The per-
pendicular projections
Lo, Lag, 4£,, and £5,
respectively, of these
directions (Figure 4)
were measured and shown
inp labile |S). a Eheise
lengths £,, £5, 43, and
44, represent £ in Equa-
tion 12 for total trans=
port calculations.

Shear Velocity U,.

Considering the

N

NW

VA
a

Fig. 4

a

Equation 10 was used to obtain U, as follows:

Ma Gelg te tee, am

a

0g Utes S 0.) ws

U - U'

(13)
Zz

6.13 Log Fr

U' and Z' were calculated using Zingg's formula and are tabulated

in Table 4.

Equation 13 was used to determine Wire

All wind-speed measurements

were made 18 feet above the sand surface, so this value was used for Z
in Equation 13 throughout the calculations.

A sample calculation is
Reach 1
dso = 1.3 mm
From Table 4,

A"
OY

ORO42 7 eit
38.00 ft/sec

Consider wind speed 27.7 mph = 40.70 ft/sec
From Equation 13

OS
Z
6p ignore
ens

= 40.70 = 38.00
18
0427

= @. Os sec/Gee

6.13 Log

The same calculations were made for all reaches, and at different wind
speeds. The calculated data are summarized in Table 5.

Total Annual Transport Q. Now having all the data required for
calculation of transport Q (Equation 12), the total annual transport
was calculated for each reach, for all wind directions contributing to
inland transport. Tables 6 through 13 show these calculations.

The total annual inland transport Q was found to be about 10,700
cubic yards per year (Table 14). It should be noted that this quantity
perhaps is on the low side since the anemometer was inoperative for many
hours during the year. It should also be noted that no reduction was
made in rate of transport for sand being in the moist condition as in-
vestigated by Belly(2)

REFERENCES

Bagnold, R. A., The Physics of Blown Sand and Desert Dunes, William
Morrow and Co., New York, 265 pages.

Belly, Pierre-Yves, Sand Movement by Winds, Univ. of California,
Inst. of Engrg, Res. Report, Series 72, Issue 7, July 1962, 90 pages.

Horikawa, K. and H. W. Chen, Sand Movement by Wind (On Characteristics
of Sand Traps), Beach Erosion Board Tech. Memo. No. 119, August 1960,.
51 pages.

Johnson, J. W., Sand Movement on Coastal Dunes, H.E.L., Univ. of
California (HEL-2-3), Jan. 1963.

Kawamura, R., Study on Sand Movement by Wind, Report of Institute
of Science and Technology, Univ. of Tokyo, Vol. 5, No. 3/4, Oct.
1951.

Kadib, A. L., Sand Transport by Wind, Studies with Sand (0.145 mm
diameter), Univ. of California, Hydr. Engrg. Laboratory, Wave
Research Pro ject (HEL-2-5), June 1963.

O'Brien, M. P., and B. D. Rindlaub, The Transportation of Sand by
Wind, Civil Engineering, Vol. 6, No. 5, May 1936, pp. 325-327.

Zingg, A. W., Wind-tunnel Studies of Movement of Sedimentary
Materials, Proc. Fifth Hydraulic Conference, State Univ. of Iowa
Studies in Engrg., Bull. 34, 1953, pp. 111-135.

Table 1

Physical Characteristics of Salmon Beach

Reach No Length along d dso Remarks
are 50 20
the Coastline (mm) D
(ft)
1 2200 1.30 2.29 Naturally graded
sand
2 1350 1.20 2.20 "
3 1700 0.380 1.24 m
4 1200 0.450 1.35 ie
5 1700 0.355 1.22 i
6 1350 0.800 -1.80 i
i 1800 0.600 1.55 ie
8 1800 0.580 1.53 i

Table 2
Duration of Wind Per Year for Different Wind Speed (Pacific Marine Station)

Duration of Wind (Hours per Years)
Speed Speed

ft/sec. mph N NW WwW SW S SE E NE

ICA et O 4 45 ok gaeeS 2 1 1

Nopoe ble 20 83 13 12 18 2 38 2

Meme 12.4 3 50 1 7 2 1 2

Ses. 2212).8 10 23 6 4 5 6 20

20. 13.6 5 24 1 1 3 2

21.02 14.3 12 39 5 5 6 7 24

278) «(14.8 1 33 4 1 1

22.9 15.6 14 35 5 5 ®) 9 10 1

2362) 115.8 28 1 1H 1

amie) 116 58 10 53 2 3 6 8

Zone 1.8 1 25 1 1

26.6 18.1 4 28 1 5 12

26. 18.8 15 1 1

20ra, 19.3 6 14 2 2 4 10 5

29 191.8 18 4

303. 20).6 14 2 1 4 5 3

32). Pl Ul 6 13 3 7 7 1

Banm2 22.6 5 8 3 3 8 4

A PSA 8 10 iL iL 2 7

36.3 24.6 2 14 3 1 2

B78) Ba av 8 3 4 2

3073 726). ¢ 8 5 3 3 1

AD OT 3 2 5 2
2628.0 5 1 2 16

Total hrs - 1135 hrs. of wind >10 mph
Calm - 3555 hrs.
No records - 4070 hrs.

Table 3

Perpendicular Projections for Different Wind Directions *

Reach Length Representing Qy Lo £3 Qa
No. Q (ft) grain dia. d50 (ft) (ft) (ft) (ft)
(mm)

1 2200 1530 1900 200 1400 1450
2 1500 1.20 1400 200 900 900
3 1700 0.38 1600 150 1150 1300
4 1200 0.45 1150 150 850 800
9) 1700 0.355 1500 400 1100 1400
6 1350 0.80 1300 300 700 1000
ia 1800 0.60 1700 500 900 1500
8 1800 0.58 1400 900 500 1700

* South, SE, E and NE winds do not contribute to inland movement

Table 4

Calculations for the Focal Point Using the Zingg Formula

Reach. No. dso Z' =10d mm Z! U' = 20dm U' ft/sec
mm (ft) (m /h)
1 1.30 13.0 0.0427 26 .00 38 .00
2 1.20 12.0 0.0394 24.0 35.00
3 0.38 3.8 0.9125 138 11.30
4 0.45 4.5 0.0147 9.0 13.20
i) 0.355 3.59 0.0116 7.10 10.40
6 0.80 8 .0 0.0262 16 .00 23.50
q 0.60 6.0 0.0197 12.00 17.60
8 0.58 5.80 0.019 11.60 17.00

Table 5

Calculation of U,, for Different Reaches and Wind Speeds

U Wn Ss Avena Sh ea ft/sec
ie +15 GuisimosiziZ!
eac eac e€ac eac eac €ac eac eac
No.1 No. e WOs8 INOot! INO. BS INGco NOo7 No. 8
14.70 - - 0.175 0.079 0.22 - - 2
16.50 - = 0.268 Ot O.elO = - =
18.20 - - 0.356 0.264 0.40 - 0.038 0.065
MES} 6 {3} - - 0.385 0.286 0.428 - 0.066 0.098
20.0 - - 0.45 0.358 0.49 - 0.130 0.164
21..02 - - 0.50 0.410 0.55 - 0.19 0.22
21.8 - - 0.54 0.450 0.58 - 0.23 0.26
22.9 - - 0.61 0.510 0.635 - 0.29 0.32
Zee - - 0.615 0.83. 64! - 0.31 0.34
24.7 - - 0.690 0.605 0.73 O.-:069 0.39 0.42
26.2 - - 0.780 O.695 0815) O67 0.48 (0) Bi
26 .6 - - 0.795 0.710 0.825 0.180 0.49 0.525
BG - - 0.840 OLWiGi0) 04838) 108235 OR55 0.58
28.4 - - 0.88 0.800 0.92 0.29 0.59 0.621
29.1 - - 0.918 0.84 0.954 0.328 0.63 0.66
30.3 - - 0.98 0.90 0.97 0.40 0.694 0.73
BA 50) - - iL (OZ 0.99 1.06 0.494 0.73 0.81
Sa) 57} - - eels L50Ha Wel Opas 0.80 0.882
34.7 - - 1S Tolar le? 0 065 0.88 0.964
3650 - 0.074 1.24 Loe eA Ooi 0) 5S) 1.05
BT ots - 0.172 1.32 120 le Omen Ones G05 Wk)
39.3 082 0.265 139 esi 1054357 0.0 hails} I ot
40.7 0.168 0.31 1.464 144 a0) ale aO 1.207 1.35
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Reach

Table 14

Total Transport per Year

Q (1b)
1,600
12,820
6,650,000
3,7 581000
8,450, 000
1,050,000
3, 600 , 000
5,400, 000
28,922,420 1bs.
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