REPORTS AND PAPERS, HYDROLOGY—1934
`
`395
`
`To meet the need for measurement of the vertical as well as the horizontal
`catch of rainfall, Pers put up ortho-and vectopluviometers, with one horizontal
`and four vertical orifices. He preferred, however, to avoid the necessary com­
`putation by cutting the top and body of the rain-gage to match the actual slopes
`and outline of that portion of a drainage-basin to be "covered" by the gage*
`Such gages are in use in the French Dauphine Alps for the Lake Blanc Basin. Un­
`fortunately, even though these gages were set up on posts about five meters
`high, no provision was made for shielding against wind.
`[19] H* Koschmieder (translated by R. J. Martin), Methods and results of definite rain-
`measurements, Mon. Weath. Rev., Washington, v. 62, pp. 5-7, 1934. Compares
`catches of rain in pit-gages against those in ordinary type on Schneekoppe. For
`the prevention of insplashing the pit-gages were surrounded with discs of brush
`having vertical bristles or with honeycombs of galvanized iron, having square
`cells 4 by 4 cm. These protective devices were 1 or 1.5 meters in diameter and
`level with the ground. The smaller were found to be essentially as effective in
`preventing insplashing as the larger, and the brush-protected gages caught about
`the same as the honeycombed. The latter type was considered preferable because
`it let blowing sand through into the pit and permitted the observer to walk out
`to the gage without taking up the splash-protector* The pit-gages caught twice
`as much as the unprotected with upslope winds of 9 m/sec and three times as
`much with 15 m/sec* The raised gages in rains of fine drops, however, caught
`more than the pit-gages. A raised gage, protected by a disc of brush parallel
`to the 8lope, caught 12 per cent more of the fine rain than the pit-gages in
`light upslope-winds and 44 per cent more in moderate. The protected pit-gages
`are useless for snow-measurement.
`
`LAMINAR SHEET-FLOW
`
`Robert E. Horton, H. R* Leach, and R. Van Vlietft^
`
`Introduction
`
`Laminar sheet-flow may be defined as the flow of a thin sheet of viscous fluid
`under conditions such that turbulence does not occur.
`
`Direct surface-runoff from the ground takes place initially as the flow of a thin
`film or sheet of water* Because some of the water is continually being absorbed by
`the soil and because of the roughness of the ground, the existence of the flowing
`sheet or film on the ground surface during rain is often overlooked or is barely
`noticeable* Studies have been made which show that the form and characteristics of
`the hydrograph of a stream are governed mainly by the phenomena of ground surface-
`runoff before the water enters definite stream-channels. To determine these charac­
`teristics a knowledge of the laws governing the flow of water in thin sheets is neces­
`sary.
`
`Surface erosion of soils results from sheet-flow of water and its occurrence is
`governed in part by the depth and velocity of overland flow during rain.
`
`The phenomenon of flow of water in thin sheets has received little attention and
`excepting the work of Jeffreys, hereafter cited, no one seems heretofore to have at­
`tempted to determine experimentally whether the Hagen-Poiseuille law is verified by
`actual flow of this type. The purpose of the experiments herein described is to fur­
`nish definite information on this point.
`
`Theory of laminar sheet-flow
`
`Let Figure 1 represent the longitudinal profile of a relatively wide channel in
`which a viscous flow Q, at depth D, is taking place, the channel being level at
`right-angle8 to its axis.
`
`'Consulting Hydraulic Engineers, Voorheesville, New York.
`
`Petitioner Walmart Inc.
`Exhibit 1035 - Page 1 of 12
`
`

`

`594
`
`TRANSACTIONS, AMERICAN GEOPHYSICAL UNION
`
`In foot-pound-second units, lets s = the slope
`of the bottom, sin a; D = the depth from the water-
`surface to the bottom (feet); d = the distance from
`the bottom to any point P within the liquid (feet);
`Q = the total discharge (cubic feet per second); q =
`the discharge per unit width, Q/width (cubic feet per
`second per foot); u = the velocity at distance d
`above the bottom (feet per second); vs = the surface-
`velocity (feet per second); v = the mean velocity,
`q/D (feet per second); p = the density (pounds per
`cubic foot;; yi = the absolute viscosity (pounds per
`second per foot); v = the kinematic viscosity (feet
`per second) = \x/p* Assume that the frictional re­
`sistance between the air and water-surface is zero.
`
`FIG. 1- LAMINAR S H E E T - F L OW
`
`For uniform flow, D = constant and there is no acceleration* The accelerating
`force due to the weight of the water above any plane PP1 parallel to the bottom is,
`therefore, equal to the frictional force acting along the plane* The accelerating
`force of the weight of the liquid per unit-area above the plane PP1 acting parallel to
`PPf is
`
`f = pg (D - d) s
`
`The frictional force due to viscous shear per unit of area on plane PP1 is
`
`-f = ft (du/dd)
`
`Equating (l) and (2)
`
`from which
`
`and
`
`p g (D - d) s = p. (du/dd)
`
`du = ( p / n) gs (D - d) dd
`
`u = (gs/\>) (Dd - d V 2) + C
`
`(1)
`
`(2)
`
`Since with viscous flow, u = 0 at the bottom, where d = 0, C is therefore zero, and
`
`.2
`u = (gs/\)) (Dd - d*/2)
`
`At the surface, d
`
`= D and
`
`vfi = (gsA) (DVO
`
`The mean velocity is
`
`v = (1/D) udd = (gs/bu) fo (Dd - d2/2) dd = (gs/bo) (Dd2/2 - iZ/6)
`
`v = (gs/u) (D2/3) = 10.72 (D2/\j) •
`
`and from equation (4)
`
`v8 = (5/2) v
`
`(S)
`
`(4)
`
`(6)
`
`(6)
`
`Also, as plane PP1 approaches the surface the accelerating foroe due to the weight of
`the liquid above PP* approaches zero, and hence at the surface equation (l) becomes
`zero and (du/dd) is zero.
`
`The discharge per unit width is
`
`q = tD = (gs/u) (D5/')
`
`(7)
`
`Petitioner Walmart Inc.
`Exhibit 1035 - Page 2 of 12
`
`

`

`REPORTS AND PAPERS, HYDROLOGY--19M
`The depth and velocity in terms of q are, from equation (7)
`1/3
`D= (3u q/ga)
`
`aDd
`
`v = qjD = (ga/3 u q) 1/3 q = (ga/3u) 1/3 q 2/3
`
`S96
`
`(8)
`
`(9)
`
`and from (6)
`
`(10)
`
`(11)
`(12)
`(13)
`(14)
`
`Vs = (3/2) (ga/3u)1/3 q2/3
`In Engliah units, for g = 32.16, equationa (8), (9), and (10) reduce to
`1/3
`D = 0.4535 (qui.) ,in feet
`1/3
`= 5.442 (qu/s) ,in inchel
`v = 2.207 (s/u)l/3 q2/3, feet per second
`1/3 2/3
`,feet per second
`va = 3.310 (a/u) q
`Experiments
`A series of experiments on Iheet-flow waa performed at the Horton Hydraulio labor­
`atory in February 1934. Owing to the oonditions prevailing at the time the experiment.
`were made they were carried only alightly beyond the limit of viacous flow.
`Experimental flume--The flume consisted of a wooden trough 6.64 inches wide and
`about 4 feet iong, the upper 13 inches of Which was used for an approach channel and
`water-supply inlet. The construction and general arrangement are shown in Figure 2.
`The bottom was made of I-inch white pine, which was prevented from warping by 1 1/2
`inch angle-iron cleats on the bottom and by angle-iron rails along both sides. It was
`
`
`finished smooth with No, 0 sandpaper but was slightly concave along the axis, being
`about 0.02 inch low at mid-length. The sides were formed by 1/4 inch by 1 ipch
`white-pine strips,' placed between the iron side-rail and the bottom-board. The top
`sdge of the side-angles was 1/2 inch above the bottom and formed the reference-base
`for all depth-measurements.
`The orifice-gate at the lower end of the approach channel was not used. A brass
`lill, 1 1/2 inches downstream from and 0.17 inch higher than the gate-sill, marked the
`beginning of the flume proper. This sill was 0.14 inch wide and projected 0.015 inch
`above the bottom. This projection did not create any visible disturbance in the
`.sasuring reach. The flow was introduced through a vertical pipe 1 1/2 inches in di­
`
`ameter by 3 inches long, at the upper end of the flume.
`An adjustable oontrol-weir at the upper end of the flume controlled the height of
`
`the water in the intake-pool. The surges due to the upward flow trom the well were
`ainimized by a defleotor which directed the flow toward the control-weir.
`Positions in the ,measuring reach were determined by means of staticn and course­
`lines drawn on the bottom with waterproof ink. Longitudinal course-lines, parallel to
`the axis, were drawn along the center line and at I, 2. and 2 1/2 inches on each side
`thereof, bringing the final lines 0.32 inch from the sides of the flume. They were
`numbered 1/2, I, 2, 3, 4, 5, and 6 1/2, from left to right. across the flume.
`Stations, in inches, were Ihown by a soale ma rked along the center line, begin­
`
`ning at the control-sill and running downstream. Transverse-lines at right-angles to
`the center line were drawn at 6-inch intervals, beginning at the control-sill.
`Positions were designated by station and course, 12-2 being a point 12 inches
`downstream from the control-sill, on course 2 (1 inch to the left of the center line).
`
`Petitioner Walmart Inc.
`Exhibit 1035 - Page 3 of 12
`
`

`

`Apri 1, 1934
`
`Sea les - as shown
`
`DETAILS OF APPARATUS
`HORTON HYDRAULIC LABORATORY
`
`EXPERIMENTS ON LAMINAR SHEET-FLOW
`
`FIG.Z
`
`Scale
`
`R|| [I
`co
`
`DETAILS OF S^ILL SECTION
`
`1
`
`Scale
`
`DETAILS OF FLUME AND OF APPARATUS FOR MEASURING DEPTH5
`
`J
`
`IR?
`
`4
`
`5.G
`
`J
`
`t-vL
`
`W
`A—i
`V
`If
`
`L
`
`W Bo
`W Bo
`FT 1^
`CP
`
`Bearinq PLATE.
`
`bearinq block
`
`^ trouqh
`Movable divertinq
`
`T
`, - m
`
`J P Friez^Evaponation
`
`Hook Gaqe
`
`"jf^
`
`5 inches
`N <fc
`
`Scale
`
`AND SECTION
`
`•34.01 ir?
`
`* * i£* arcqle side nail
`
`is.
`
`Overflow waste JO
`
`trouqh
`
`overflow
`/Adjustable ~.
`
`Level »r?q
`
`screw
`
`o
`
`Petitioner Walmart Inc.
`Exhibit 1035 - Page 4 of 12
`
`

`

`REPORTS AND PAPERS, HYDROLOGY—1934
`
`397
`
`Measurement of depth—The depth was measured by a Fries micrometer hook-gage,
`mounted as shown on Figure 2. The sliding supports enabled the gage to reach any part
`of the flume downstream from Station 3.2.
`
`The micrometer-scale on the gage was graduated in 0.001 inoh. The depth of water
`was determined by the difference between the gage-reading at the bottom of the flume
`and the gage-reading at water-surface, using in both cases the bottom of the curved
`.hook. Readings on the bottom were made by placing a flat brass strip 3/8 by 1 inch on
`the bottom and slowly lowering the gage until a piece of bond paper plaoed between the
`bottom of the gage and top of the brass plate could just be moved. The gage-reading
`was corrected for the thickness of the brass plate and paper (0.062 inch; to get the
`true bottom-reading.
`
`Measurements of water-surface were made by slowly lowering the gage until the
`bottom of the hook touched the water-surface. This contact was distinct and definite,
`at the instant it was made there was a definite capillary rise on the hook-gage. The
`hook was wiped dry after each measurement. Repeated measurements rarely differed by
`sore than 0.002 inch and generally by less than 0.001 inch.
`
`Slope—The slope of the side-rail was measured by means of a U-gage. It con-
`slsted of two glass tubes, 0.4-inch (internal) diameter by 7 inches long, each mounted
`on a block and connected by a rubber tube 4 feet long. The gage-tubes were placed 30
`inches apart on the side-rails and readings of the water-level in both tubes were tak­
`en. The positions of the tubes were then reversed and the readings repeated. The
`difference in level along the side-rail was obtained from the average difference of
`the A-reading8 and the B-readings. The water-level in the tubes could be read to l/40
`Inoh, equal to a slope of 0.0008 in 30 inches and the probable error in slope is about
`0.0004. The effect of this possible error is mainly in the flat slopes.
`
`The hook-gage readings to the bottom of the flume were taken with reference to
`the side-rails. The slope of the water-surface was obtained by adding algebraically
`the difference between the water-surface readings at stations 12 and 50 (stations 12
`and 24 for runs 1 to 3) to the slope of the channel-bottom.
`
`Volume of outflow—A movable trough was arranged at the downstream end of the
`flume by means of which the flow through the flume could be instantly diverted into or
`*ut of a 25-galIon can. With steady flow established, the discharge through the flume
`was caught in the can for a period of 10 to 30 minutes. The outflow was weighed on a
`Fairbanks silk scale, capable of weighing to 1/4 ounce. The discharge in a given ex­
`periment usually weighed from 100 to 200 pounds. Under these conditions extreme ac­
`curacy in the determination of the volume of outflow was attainable.
`
`Measurement of surface-velocity—The surface-velocity was determined by the time
`of transit of the front of a patch of color over the 24 inches between stations 6 and
`50. After trying .fluorescein and various other materials, ordinary washing-bluing was
`adopted. This gave a distinct, easily observed color-patch. The bluing was dropped
`on to the surface near Station 2, from the needle of a hypodermic syringe, held about
`0.1 inch above the water-surface. Observations were made with a stop-watch graduated
`in 1/5 seconds.
`
`Several comparisons made with circular paper-floats, 1/8 inch in diameter, dropped
`on the surface from the point of a knife, gave velocities in good agreement with the
`front of the color-patch, as the following tabulation shows.
`Time in seconds
`Color
`Paper
`
`
`
`Course No. Course No.
`
`3 (center line)
`
`5 (2 inohes right of center line)
`
`5.5 (0.52 inch from side)
`
`5.4
`5.4
`5.4
`5.0
`4.9
`4.9
`4.8
`6.0
`5.0
`
`5.6
`5.4
`5.4
`5.0
`5.0
`5.0
`4.8
`4.8
`4*8
`
`Petitioner Walmart Inc.
`Exhibit 1035 - Page 5 of 12
`
`

`

`598
`
`TRANSACTIONS. AMERICAN GEOPHYSICAL UNION
`
`Prom 15 to 50 velocity-observations were made for each run. With a probable er­
`ror of 0.2 second in observed time, single observations were subject to an error of 20
`per cent in the case of the highest velocities, decreasing to 4 per cent for the low­
`est velocities. The error in mean velocity for a run probably did not exceed 5 per
`cent in any case.
`
`Experimental procedure and observations—Before turning water into the flume for
`a series of runs, hook-gage readings were taken on the bottom, generally at all points
`where depths were to be read. The amount of swelling of the wooden flume was deter­
`mined by repeating these measurements at the end of a series after the water had been
`turned out of the flume* The swelling did not exceed 0.008 inch at any point and usu­
`ally averaged less than 0.005 inch. The observations indicated that all or nearly all
`of the swelling occurred during the first and second runs.
`
`The flume was then leveled transversely and the longitudinal slope measured. The
`flow was started and the end-weir adjusted so that there was a slight flow over the
`weir. At a given time the outflow was diverted into the measuring can and readings of
`the water-surface were taken at 6-inch Intervals along the center line, beginning with
`Station 6, together with readings at Station 5.2 and 54.0. In addition, readings on
`courses 1 and 5 were generally taken at stations 12, 18, and SO to determine the
`transverse-variation in depth.
`
`Velocity-observations were then made, taking three readings (frequently more and
`occasionally only two) along each of the five courses. Temperature-readings of the
`water were taken from time to time in the inflow-pool during the course of the run.
`There was generally little or no change in the temperature of the water. When read­
`ings were completed, diversion into the can was stopped and the water was weighed.
`
`With the lower flows the surface was smooth. At the higher flows stationary rip­
`ples projected diagonally downstream from each side of the flume and in certain cases
`cross-ripples projecting both upstream and downstream were formed*
`
`Attempts were made to obtain color stream-lines by introducing a steady flow of
`color from the syringe-needle projecting below the surface. At the lower velocities
`some of these were fairly successful and produced thin threads of color of uniform
`size and intensity* The path was generally more or less sinuous, although the thread
`was unbroken and of uniform size throughout its length. At high velocities, turbulenoi
`at the needle made it difficult to obtain a thin thread and the color-stream appeared
`as a broad, sinuous band*
`
`Excluding preliminary trials, 28 runs were made, in the first four of which the
`flow contained more or less entrained air*
`
`Table 1 gives the observed or experimental data for each run* Columns 5 to 9,
`inolusive, give the average of all of the depths, in inches, measured at different
`points on the section at the designated station* Column 10 gives the arithmetic aver­
`age of the mean depths at stations 12, 18, 24, and 50* Columns 11 to 15, inclusive,
`give the measured surface-velocities on five lines or courses at distances from the
`left-hand side of the flume, as indicated in the column-headings* Observations taken
`on two additional courses, one inch each side of the center line, as a check on the
`center-line velocity, gave velocities nearly the same as those observed at the center
`line. The average surface-velocity as given in column 16 was obtained from the mean
`velocities of all seven courses, including the two adjacent to the center line, but
`giving the observations for the side-courses (0.5 and 5.5) half the weight of the
`other observations.
`
`The kinematic viscosities given in column 19 in foot-pound-second units are based
`on the following data:
`
`Temperature
`°F
`50
`40
`50
`
`0.0000200
`0.0000167
`0.0000141
`
`Temperature
`
`60
`70
`80
`
`0.0000121
`0.0000106
`0.0000095
`
`Petitioner Walmart Inc.
`Exhibit 1035 - Page 6 of 12
`
`

`

`REPORTS AMD PAPERS, HYDROLOGY— 1934
`
`S99
`
`TABLE I - EXPERIMENTAL DATA, LAMINAR SHEET- FLOW
`
`S l o pe
`3 to Si to 30
`Water-
`Surface
`
`Measured Depth
`Incpga,
`sfntibn
`
`07©5
`O950
`.0993
`.1027
`.1047
`
`J097O
`1060
`
`.0104
`OII4
`0115
`00533
`.OOS62
`.00667
`XX>2&4
`00375
`•0O2O3
`.0O39C
`.00357
`
`fceraar Measured Surfoce-teJocity Mean Orsdnqi
`r t Der Sec
`Deott)
`Surface
`5 £5
`pa­ Wafer ftV
`i 3
`.5
`2* Gaiter 2" 2i* 2 So rt. wAn •F ftV
`inches
`e V
`Uft Line RKjht Ok
`r>
`114) OS) 06)
`ou
`<«y OS
`us;
`ua>
`booo
`. 5 55 .500 527 . 5 88 £25 .543 O0249 41.9 + I6IS
`. o ea
`.0971 740 £ 00 .769 .833 909 .804 .00236, 43.4 1575
`.1045 .930 .90S 3 30 sns J.0OO .951 O0S33 4S.4 1520
`1583 852 £ 52 330 908 3 08 .908 .00824 43.4 1575
`.1378 .897 .922 858 £97 £69 £92 XXXS4 32.6 19KD
`.1312 672 .714 .672, .669 .697 .689 . 0 O 4 87
`.1245
`1258 S©3 588 .560 360 .601 569 .00386
`.1135
`.1164 .392 .362 -370 .406 .406 381 .0O242
`IS65
`.1357 833 .823 .870 .857 £80 .646 .00751
`.1446 .832 £38 .832 £32 £32 .838 .00615 32.8 1305
`OOS48
`1460
`1460 1463
`1420 I
`.1690
`.00553
`ITOO 1723
`1690 I 0 0O
`.9&S 1000 i.OOO LOOO .995 .00966
`1770 .1647
`.0106.
`• i486 I2SO 1177 JI30 iise 1.19© 1.163 .00984 32.6. I9IO
`1487
`1620 .1507
`.OIOO
`• 1343
`1430.1340
`.1346 1070 1.053 .966 1.036 LOOO 1.015 .00603 ••
`.0097
`.1093
`1043
`.1074 £IO .790 •714 .769 £ 33
`7 59 X3045S 32.8 1905
`.1050
`•0O88
`.0673
`.0830
`0634 .551 555 .457
`.504 .00245 33.4 1805
`.0810
`.S4S
`.517
`.0817
`.0770 0780
`iO\ 15
`.0822 .586 .53? .476 .545 &0O .523 .00243 3 33
`1090
`.1077
`.OI2I
`1080 .1053
`• I063 .908 .632 .790 £ 43
`.840 0055& 32© 1905
`.890
`.0124
`1300
`1293
`.1340
`.1290
`1306 1.1 II 1.156 1037 I.I5C I.I II
`326. 1910
`I.IZ1
`.OI28
`.1423
`1363
`.1483
`1360
`1407 1.212 1137 1.191 1.131 1.228 1.187 .OIOI2.
`1185 1110 1.000 1.000 1.000 1.000 IOJO XW747 523 1363
`.1190
`.1160
`I I 70
`.1500
`I4S0
`.1430
`.1480 1.177 1250 I2SO I2SO 1250 1.244 .01400 357 1808
`.I8IO
`1690
`.1710
`• I77S 1 370 1540 1540 I430 I2SO 1 4 47 .OI577 3 55
`kOlS
`.1550
`1450
`1470
`.1528 1539 1667 1539 U30 1.430 1.543 .01938 36.6 1780
`1640
`.1270
`II40
`.1140
`.13 IO
`.1215 I.90S 2 2 20 2000 — — 1980 .OI641 35.1 1628
`OfJSO
`0750
`0730
`.0770 .Q763
`.0748 1.430 1250 I.2SO 1.430 1.430 1.340 .0C6I7 3 5.S
`IftlS
`.0810
`.0750
`.0860 0780
`0 7 9O
`0762 1540 1.430 1.430 1.430 1670 1.400 .00716 36.0 1600
`.U40
`102O
`1040
`.1095 2.000 2.220 2.000 2 0 00 2.000 2037 01530 *4.9 1835
`XI730
`O7I0
`OTOO 1.670 1430 1250 1.430 ».S40 1383 .00663 35.4 1620
`
`1220
`.1540
`
`.0125
`.OI25
`.OI3I
`OI9G
`. 0 2 04
`.0358
`.03CS
`0 3 55
`0 3 5S
`0 4 74
`0 4 71
`
`TABLE 2-RESULTS OF EXPERIMENTS
`
`
`
`Hear? Raiio Qilculaksd Calculated Reynolds Qilculaksd
`Mear?
`Mear? Slope
`
`
`of CoUt) Velocity Velocity Velocity Number Type
`Depth Of Kinematic Surface-
`NO. ptatttoao Water Vbcoiify Velocity
`of
`to Laminar 1&\
`Col. (s) flow flow yji now
`feet Surface
`Sta.vzio30
`irt
`0
`u-t
`s
`W) (6> a) (8) (9»
`w
`(10;
`(»}
`(9)
`. 3 43
`.362
`. 3 65
`672,
`. 3 64
`.292
`>4l4
`. 3 63
`. 8 04
`.536
`. 6 12
`. 6 44
`.623
`. 6 34
`.951
`
`X
`'D
`
`V
`#>
`
`00663
`0 0 8 09
`.00671
`
`. 0 1 04
`
`• O I I5
`
`.00001615
`157S
`1520
`
`10
`
`1
`2
`3
`
`3 20
`.509
`.617
`
`6 06
`. 7 09
`.747
`
`(«)
`154 Laminar
`«
`I SO
`3 SO
`
`4 .01322
`
`.00539 .00001575
`
`. 9 08
`
`.606
`
`.623
`
`.G/4
`
`. 6 86
`
`.641
`
`.676
`
`5 23 Lamifwr
`
`5 .01315
`.00562. .00001910
`6 .01093 OOZ64
`' 7
`.01048 J00203
`8
`0 0 9 07
`.00396
`9
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`
`(.«>' urbolor f R o*
`
`Petitioner Walmart Inc.
`Exhibit 1035 - Page 7 of 12
`
`

`

`400
`
`TRANSACTIONS* AMERICAN GEOPHYSICAL UNION
`
`Laminar Flow
`
`"Turbulent Flow(Manninq)
`
`IG5.I cfar§" for n-aoos
`-
`Horton's criterion
`
`for n*aoo9
`
`Calculated Velocity for Laminar Flow, Ft per Sec
`
`F I G .3- C O M P A R I S ON OF O B S E R V ED AND C A L C U L A T ED
`VELOCITIES FOR L A M I N AR AND TURBULENT FLOW
`
`Results of the experiments
`
`Table 2 gives the results of the experiments with the depth (column 2) expressed
`in feet. From equation ( 6) it will be seen that for laminar sheet-flow the mean
`velocity is two-thirds of the surface-velocity. Since the discharge q per foot of
`width and the depth D are known, the experimental data afford two methods of deter­
`mining the mean velocity where the flow was laminar. The mean velocities, as deter­
`mined, respectively, from the surface-velocity and from the discharge and depth, are
`given in columns 6 and 7. Since these two results appear to be about equally reli­
`able, the average of the two, as given in column 8, has been used in cases where the
`flow was definitely non-turbulent. Experience in connection with turbulent flow in
`open channels shows that the ratio of mean velocity to surface-velocity is higher than
`for laminar flow. For turbulent flow the ratio is usually 0.75 to 0.90. The ratio of
`the mean velocity, determined from discharge and depth, column 7, to the surface-
`velocity,, column 5, is given in column 9. These ratios afford a criterion for deter­
`mining in which cases the flow became turbulent. It will be noted that for experi­
`ments 1 to 20, inclusive, the ratio of the surface-velocity to the mean velocity is
`always less than 0.75 and,with a few exceptions, close to two-thirds, indicating that
`in all these experiments the flow was distinctly laminar. Also it will be noted that
`with the exception of experiments 6 and 7, the calculated velocity for laminar flow,
`as given in column 10, is generally in good agreement with the observed mean velocity,
`given in column 8.
`
`Column 11 of Table 2 gives the calculated velocity for turbulent flow, assuming
`a value of Manning's roughness coefficient n = 0.009.
`
`Figure 3 shows by black dots the observed mean velocities plotted in terms of the
`calculated velocities for laminar flow. For the first 20 experiments, excepting num­
`bers 6 and 7, the plotted points generally fall close to the line of equality. Crosses
`plotted on the same diagram show the velocities as they would be for turbulent flow.
`
`Petitioner Walmart Inc.
`Exhibit 1035 - Page 8 of 12
`
`

`

`REPORTS AND PAPERS, HYDROLOGY—1954
`
`401
`
`For reasons subsequently explained, the velocity of turbulent flow, if auch flow oould
`' ooour at very slight depths, would be greater than the velocity of laminar flow. It
`; will be noted that for velocities less than one foot per second the calculated veloci­
`ties for turbulent flow are above the line of equality. They are also generally higher
`\ than the observed velocities. For experiments 6 and 7 the calculated velocities for
`» turbulent flow and the observed velocities are in agreement. This of itself would
`• indicate that in these cases the flow was turbulent. A similar result might, however,
`be obtained by an error in the measurement of slope. [Reduction of one-half in the
`' measured slope in experiments 6 and 7 would bring these points into harmony with the
`I other data and would also bring them below the calculated velocities for turbulent flow
`at the corrected slope.] At the time the experiments were made and in advance of the
`calculations, it was noted that the measured slope for these two experiments was ap­
`parently in error. Both the ratio of mean velocity to surface-velocity and the appear­
`ance of the water-surface during the experiments indicate that the flow was, in fact,
`distinctly laminar in these two cases.
`
`With reference to experiments 20 to 28, inclusive, several of these are for
`velocities close to unity. In this region some fall above and some below the line of
`equality. Lines added to the diagram connecting the observed velocities with the cal­
`culated velocities for turbulent flow show that in every instance where the calculated
`velocity is close to but below one foot per second, the observed velocity is materially
`below the velocity for turbulent flow. It appears reasonably certain that in every
`instance where the observed velocity was less than one foot per second the flow was
`definitely laminar.
`
`In case of experiments numbers 21 to 24, inclusive, and number 27, the observed
`velocities were materially in excess of one foot per second. In all these cases the
`ratio of surface to mean velocity equals or exceeds 0.73 and in all these cases the ob­
`served mean velocity is in good agreement with the calculated velocity for turbulent
`flow.
`
`For conditions where the flow is definitely laminar, the mean velocity for sheet-
`flow, as given by equation (5), is confirmed by the experiments and this equation,
`which has heretofore had little more than a theoretical basis, can apparently be ac­
`cepted as reliable for determining the velocities of laminar sheet-flow on smooth
`surfaces. Theoretical considerations indicate that the velocity of laminar flow in
`accordance with Poiseuille's law is independent of the roughness of the surface.
`Whether this is or is not true in fact remains to be determined from additional exper­
`iments.
`
`Reynolds' criterion
`
`The general expression for the resistance to the flow of fluids in pipes is
`
`R = pv2 f (vd/u)
`
`(15)
`
`in which (vd/o) is the Reynolds number. In a wide channel the linear dimension repre­
`sented by the diameter, d, of the pipe may be replaced by the depth of water, D, and
`equation (15) becomes
`
`or, since vD = q
`
`R = pv2 f (vD/o)
`
`R = pv2 f (q/u)
`
`(16)
`
`(17)
`
`Jeffreys (Harold Jeffreys, The flow of water in an inclined channel of rectangular
`section, Phil. Mag., s. 6, v. 49, No. 293, London, May 1925) has suggested that (q/^>)
`be called the Reynolds number, since both q and t) are under direct control. Each form
`has advantages, depending on the problem.
`
`In steady flow, with no acceleration, R per unit-area of channel is measured by
`the component of the weight of water parallel to the bottom of the channel. Then R =
`Pg DS, and, from equation (16)
`
`(gDs/v2) = f (vD/v)
`
`(18)
`
`Petitioner Walmart Inc.
`Exhibit 1035 - Page 9 of 12
`
`

`

`402
`
`TRANSACTIONS, AMERICAN GEOPHYSICAL UNION
`
`.OI
`
`_
`
`• - Viscous Floy*/
`x-Turbulent Flow
`v-Mcan velocity,ft p er sec
`D-Depth, feet
`v - Kinematic viscosity, ft2/scc -
`5 - slope of water-surface
`
`Reynolds Number, N'^/u
`
`2000
`
`0001
`3000
`
`FIG.4-REYN0LD5 NUMBER FOR SHEET-FLOW EXPERIMENTS
`and as v2 = (q2/b2)
`
`(gD3s/q2) = f (YD/d)
`
`(19)
`
`When the flow is viscous, f (vD/o) can be determined from equation (5), which,
`when both sides are multiplied by D2v, becomes D2v2 = (D5gs/3) (vD/o), which is equiva­
`lent to
`
`(3q2/b3gs) = ( Y D/ o)
`
`(20)
`
`<*
`n
`Therefore as long as the flow is viscous, (D s/q ) is inversely proportional to the
`Reynolds number, (vD/v).
`
`Values of Reynolds number for the authors' experiments are given in column 12,
`Table 2. The observed values of (D3s/q2) and (vD/o) are plotted on Figure 4. The
`line shown is the graph of equation (20), which reduces to (Dss/q2) = 0.0933/fo, in
`which N is the Reynolds number, (vD/o) or (q/u).
`
`The experiments in which the flow was viscous follow the line of the equation
`rather closely but in general somewhat below it. This is due in part, at least, to
`side-effect. In runs 25 and 26 (Reynolds number 340 and 395) the capillary rise on
`the sides was noticeably greater than in most of the runs and the velocity for a dis­
`tance between 0.2 and 0.3 inoh out from the side was much higher than that of the main \
`sheet. Ac a result an unduly large proportion of the flow travelled close to the
`
`Petitioner Walmart Inc.
`Exhibit 1035 - Page 10 of 12
`
`

`

`REPORTS AND PAPERS, HYDROLOGY--1934
`
`403
`
`aides, reducing the depth in the body of the sheet below what it would have been with
`a wider channel and the same discharge per foot of width.
`
`Jeffreys found that for sheet-flow, turbulence set in when the Reynolds number
`was about 310 and states that Hopf obtained critical values between 300 and 330. Re­
`f.rring to column 12, Table 2, it will be noted that the highest value ot Reynolds
`number in any ot the authors' experiments in which the flow was definitely laminar i.
`648. The lowest value ot Reynolds number in any ot the experiments in which the flow
`was turbulent is 773. For reasons given in the next paragraph, the authors believe
`that Reynolds number, taken by itself, is not a reliable criterion of the point at
`which flow changes fram a laminar to a turbulent regime or vice versa.
`
`Horton's criterion
`
`Osborne Reynolds, as a result ot experiments on pipes, conoluded that there are
`two limits between whioh the flow in pipes may be either laminar or turbulent, these
`limits depending only on the pipe-diameter and the visoosity of the fluid. There is
`laae question as to the applicability of Reynolds' criteria for pipes in the oase of
`open ohannels. Furthermore, the pipes on which Reynolds experimented were relatively
`Booth and there is a question as to the validity of his results with reference to
`flow over rougher surfaces.
`The velooity of laminar sheet-flow varies as the square ot the depth D. The
`ftlocity of turbulent flow in a wide ohannel varies as the two-thirds power ot the
`depth. The squares of small numbers are still smaller numbers, while the two-thirds
`power of a small number is a larger number. It follows that at v