Drag measurements in roughness arrays of varying density and distribution

AgriculturalMeteorology - Elsevier Publishing Company, A m s t e r d a m - Printed in The Netherlands

DRAG

MEASUREMENTS

DENSITY

AND

IN

ROUGHNESS

ARRAYS

OF

VARYING

DISTRIBUTION

J. K. M A R S H A L L

C.S.LR.O., Division of Plant Industry, Deniliquin, N~S. W. (Australia) (Received June 11, 1970)

ABSTRACT

MARSHALL,J. K., 1971. Drag measurements in roughness arrays of varying density and distribution. Agr. Meteorol., 8: 269-292. The partition of the total shearing stress, z, between that contribution attributable to the roughness elements projecting from a wind tunnel wall, W,/F, and the shearing stress at the intervening wall surface, zg, is expressed as:

= Wr/F + ~o F'/F, where F i s the total wall area of which F ' is not covered by roughness elements. This partition of shearing stress was examined for cylinders of diameter/height ratio d/H of 0.5-5, as well as for hemispheres, forming homogeneous arrays ranging in distanceapart/height ratio from 1 to 59. Both regular and random distributions of the elements were examined. Total shearing stress does not appear to differ appreciably with the type of distribution when arrays of the same element density are compared. For widely separated roughness elements (lateral cover Lc -~ 0) which are regularly distributed the individual element drag coefficient C! referred to the freestream velocity approaches a constant value C¢o corresponding to that of an unobstructed element. The variation of Cyo with d/H corresponds generally with TILLMANN'S (1953) results. When the elements are closely spaced (Lo -~ Lc max.) so that zg is negligible, Cf varies linearly with 1/WLc and is largely independent of d/H and of element form. The points at which the mean unobstructed drag coefficients C:o of different roughness elements intersect the line relating C¢ to l/~/Lc are shown to provide an approximate indication, for the regular arrays used, of the values of l/WL, at which T# becomes negligible. At and beyond this condition of negligible zg, the wind profile parameters, the friction velocity, u., and the wind velocity, uz, at a given height, z, are related by definition to lateral cover, Lc, and the mean element drag coefficient, C:, thus:

C/L~ = 2u.2/uz 2 The above relationship linking CIL, and the wind profile parameters allows the results from this study to be compared with the field study of KtrrZaACH (1961). Comparison with the water tunnel study of SCHLICHTINC (1936) is also made. Both comparisons are relevant, the first because the eventual aim of this study is a field application, the second to include the performance of roughness element forms not used in this study. The results of the three studies are used to derive a combined relationship describing the behaviour of C: with 1/VLc and to compare the effectiveness of arrays of different element forms in reducing zu to a negligible amount. The results are discussed in relation to the problem of soil erosion by wind from partly vegetated surfaces and, in particular, the role of the vegetation elements in protecting the intervening surface.

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270

J.K. MARSHALL

INTRODUCTION

Determination of the erodibility by wind of soil from partly vegetated surfaces is hampered by the lack of information on the influence of the vegetation on airflow. Once the force at the ground surface is known, the erodibility of a particular soil can be predicted (BAGNOLD, 1941; CHEPIL and WOODRUFF, 1963). However, for partly vegetated surfaces it is necessary to relate the ground surface force to one or more parameters of the wind profile through measurable attributes of the vegetation or roughness elements. The relationship is examined in this, and a companion paper. In general, the problem is one of the effect of arrays of roughness elements of varying density and distribution on airflow, and eslzecially on the partition of the forces on the elements and on the intervening surface. This problem has many important applications, for example in aeronautical and nautical engineering where the resistance to the passage of roughened surfaces through air and water can significantly reduce the velocity of a vessel (HOERNER, 1965; SCHLICHTING, 1968). An approach to the present problem is suggested by SCHLICHTmG (1936) who examined experimentally the problem of resistance created by roughnesses on ship hulls. He considered in his eq. 15 (eq.1 here) the partition of the total resistance between that due to the roughnesses and that due to the intervening surface. While the interest here lies more in the behaviour of the second force as the roughness of a surface is increased, the approach remains valid. This paper presents the results of an experimental study in which the relationships between airflow and artificial roughness elements modelling individual vegetation elements were explored in a wind tunnel. A second paper (WooD1NG et al., 1971 ) develops further relationships contained in the experimental findings. LIST OF SYMBOLS A N D T H E O R Y

Roughness geometry and distribution (see also Fig. 1) roughness element diameter (cm) d D average distance between roughness elements (cm) height of roughness elements (cm) H total test-floor area (cm 2) F total uncovered floor area = F-Nnd2/4 (cm 2) F' A specific area (regular arrays), average floor area per element (random arrays) = FIN (cm 2) A' specific area or average floor area per element less element projected area = F'/N (cm 2) Z a frontal (silhouette) area of roughness element (cm 2) Lc lateral cover = La/A (dimensionless) Agr. Meteorol., 8 ( 1971) 269-292

271

DRAG MEASUREMENTS IN ROUGHNESS ARRAYS

number of roughness elements in array projected area of roughness element on horizontal plane (cm 2)

N Pa

Airflow h u Uh Uz U,

=

U,g T "C45o

Tg w Wr

wg x, y, z

Co CI Cfo

c~ P Re W

w~ w, P zo

freestream height (cm) mean horizontal velocity (cm/sec) freestream velocity (cm/sec) velocity at height z (cm/sec) friction velocity (T/p) 1/2 (cm/sec) uncovered floor friction velocity = (%/p)1/2 (cm/sec) total shearing stress (dynes/cm 2) total shearing stress for regularly distributed arrays aligned at 45 ° to direction of flow (dynes/cm 2) shearing stress at intervening surface (dynes/cm2) total drag force on area, .4 (dynes) drag force on individual roughness element (dynes) drag force on area, `4' (dynes) co-ordinate system: axes horizontal and parallel, horizontal and normal, and vertical and normal to flow direction overall drag coefficient = 2u2/u2 (dimensionless) roughness element drag coefficient = 2 W,/Npu2La (dimensionless) unobstructed roughness element drag coefficient (dimensionless) uncovered floor drag coefficient = -gu2,g/u~/" 2 (dimensionless) static pressure (dynes/cm 2) Reynolds' Number = Huz/v (dimensionless) total drag force on roughened surface of area F (dynes) sum of drag force on all roughness elements = ZNw, (dynes) drag force on uncovered floor area, F' (dynes) air density (g/cm 3) kinematic viscosity of air (cm2/sec) roughness parameter (cm)

Drag partition theory SCHLICHTING(1936) proposed that the overall force imparted to a roughened surface by a fluid passing over it, W, be partitioned into the force exerted on the roughness elements, W. and the force exerted on the intervening wall surface, Wg, thus: W = W, + Wg (1) Eq.1 can be rewritten in the form of a partition of shearing stresses, thus:

W/F = W,/F + Wg/F'. F'/F i.e.: Agr. Meteorol., 8 (1971) 269-292

272

J.K. MARSHALL = w, I r +

~. ~'1~

(2)

Now, by definition, the drag per unit area due to the roughness elements in eq.2 is:

WdF = W,/NA = Cs P,,~=Lo/2A putting L c = L,JA and r = pu, 2 etc., and multiplying through by

2u2,/u 2 = Cs Lc + 2ug F'/F ,~

2/pUz2: (3)

In drag coefficient notation, eq.3 becomes: C o = Cf

Lc + C~ F'/F

(4)

If zg is negligible compared with W/F, then: Co = 2u2,/u~ = CsL¢

(5)

Eq.5 defines Cy for this condition and it is possible, from data which follow, to determine its dependence upon the density of elements, as characterised by L~, in the regime where % is negligible.

METHODS A N D MATERIALS

Wind tunnel The wind tunnel used was a modified NPL-type open-circuit tunnel with an initial contraction ratio of 4 : 1 leading to the first test section. This was followed by a second contraction section of ratio 1.81 : 1 and another test section beyond which a diffuser leads to an axial exhaust fan. The tunnel is at the C.S.I.R.O. Division of Meteorological Physics, Aspendale, Victoria (C.S.I.R.O., 1967). Velocities in the first test section, which was used throughout this study, are continuously controllable in the range 2-20 m/sec. The test section dimensions are x = 183 cm, y = 124 cm and z = 69 cm. Maximum variation in velocity across the section at x = 133 cm is + 1 ~ (E. L. Deacon, personal communication). In the absence of any roughness elements, the turbulent boundary layer thickness at x = 180 cm was 4.6 cm for a freestream velocity of 20.3 m/sec. Elements and arrays Table I lists possible varieties of roughened surfaces which are of interest in studies of drag partition. In the present study, variations of roughness element density (i); size or shape (2); form (3); horizontal distribution (5b) and orientation (6) were examined. Symbols used to describe the roughness elements and arrays have been listed in the preceding section. Roughness elements of a uniform height, 2.54 cm, were used, all elements being solid, smooth wood with 2-3 coats of clear polyurethane "varnish". The Agr. Meteorol., 8 (1971) 269-292

DRAG MEASUREMENTS IN ROUGHNESS ARRAYS

273

TABLE I VARIATIONS IN ROUGHENED SURFACES

Variable

Possible variations

1 2 3 4 5

number of elements/unit area diameter, height and their ratio sphere, hemisphere, cone, cylinder, pyramid etc. homogeneous; heterogeneous forms, sizes and shapes

Density Size and shape Form Composition Distribution (a) in vertical (b) in horizontal 6 Orientation 7 Permeability 8 Surface finish

varying H regular, random, contagious variation relative to wind direction ratio gap area/frontal area viewed in silhouette smooth, roughened, with or without salient edges

¢

<

OR

)

D+d

Fig.1. Roughness array showing descriptive symbols used (see also section "List of symbols and theory"). diameter/height, d/H, ratios o f the cylindrical elements were 0.5, 1, 2, 3, a n d 5 while the hemispherical elements were o f d/H = 2. D i a m e t e r / h e i g h t ratios between 1 a n d 3 are c o m m o n to m a n y w o o d y perennial plants while cylindrical a n d hemispherical forms occur frequently. The element-spacing/height, D/H, ratios o f the roughness arrays varied from 1 to 59. In the semi-arid, s h r u b - d o m i n a t e d c o m m u n i t i e s to which the results

Agr. Meteorol., 8 (1971) 269-292

274

J, K. MARSHALL

are to be applied the shrubs are rarely more closely spaced than an average D / H of 2.

Two distribution patterns were examined in detail. Arrays of elements set out in a regular pattern (Fig.l, 2A) with the elements at the intersections of a square grid orientated parallel to the flow were examined to establish reference

Fig.2. Photograph of roughness elements, d/H -- 1, at spacing, D -- 2 H, distributed (A) regularly and (B) at random. Agr. MeteoroL, 8 (1971) 269-292

DRAG MEASUREMENTS IN ROUGHNESS ARRAYS

275

d a t a for c o m p a r i s o n with other, less easily r e p r o d u c i b l e patterns. A v a r i a t i o n o f this p a t t e r n in which the a r r a y was o r i e n t a t e d t h r o u g h 45 o was briefly examined. T h e s e c o n d d i s t r i b u t i o n p a t t e r n consisted o f r a n d o m l y a r r a n g e d elements (Fig.2B) to simulate m o r e closely the r a n d o m a n d c l u m p e d distributions typical o f n a t u r a l vegetation (GREIG-SMITH, 1964; ANDERSON, 1967). The detail in which the possible c o m b i n a t i o n s o f a r r a y c o m p o s i t i o n a n d d i s t r i b u t i o n has been e x a m i n e d is s u m m a r i s e d in Table II. TABLE II SUMMARY OF S I T U A T I O N S EXAMINED

d/H Regular distribution

D/H

0.5

1

1

2 3 4 5 6-7 8-9 10-11 12-13 14-15 16-17 18-19 20-24 25-29 30-34 35-39 40-44 45-49 50-59 Unobstructed wall . . . .

2

3

1,3

1-3

1-3 1-3 3

5

2 hs

1, 3.3

1,3 2

1,2

1,3 3 1-3 1-3 23 2,7 2 3 3 3 3 3 2, 3 2,3 2,3 2, 3 3_ 3 3

2 2

1,3 3

2

2

2,3 2, 3 2,3

3 3 3

2 2 2

2 2 2 2

2,3

2,3-

2,3 3 3 2,3 3

2

3 3

3 2,3 2,3

3 2,3

2, 3 2,3 2,3 2,3 2,3 3 3 3 3 3

3

.

.

.

.

.

.

.

.

.

.

.

2

3

5

2

1-3

2,3

.

1

1-3

3

2, 3 2,3

.

0.5

2 2

2, 3

.

d/H Random distribution

2

2 hs

2 2

2 2 2 2 2 2

2 2

2

2 2 2

2 2 2 2

2 2

2

2

2

2

2

2

2

2

2

2 2 2 2 2 2

2 2

1,2

1 indicates measurement by the momentum integral method; 2 indicates measurement by the drag base method, 3 indicates measurements of wr; and 3 indicates estimates of wr based on the value of mean unot~structed drag coefficient, hs = hemispheres.

Scaling The m a i n aspect o f scaling c o n s i d e r e d was R e y n o l d s ' N u m b e r similarity with the field situation. In A u s t r a l i a n semi-arid, s h r u b communities, the d o m i n a t i n g roughness elements are shrubs such as saltbush (Atriplex vesicaria) a n d c o t t o n bush (Kochia aphylla) a b o u t 40 c m high, o r bluebushes (Kochia pyramidata; K. sedifolia) a b o u t Agr. Meteorol., 8 (1971) 269-292

276

J.K. MARSHALL

90 cm high. Defining a Reynolds' Number using shrub height, H, and a wind-speed of 3 m/sec at twice shrub height (z = 2H) then Re = 8.1 • 104 and 1.87. 105 respectively for the two species. In the wind tunnel, at the maximum velocity used, 20.3 m/sec (uh), and roughness element height of 2.54 cm an analogous Reynolds' Number at 2 H within the boundary layer at x = 133 cm is about 3 • 104. Wherever possible, the wind tunnel experiments were repeated at lower velocities to provide data over the range 1.5-3 • 104. The experimentally determined drag coefficients were comparatively insensitive to Reynolds' Number over the range examined (Fig.3). This fact, along with the known relative constancy for cylinders over the range l 0 3 _< Re < 2 • 105, (here referred to diameter (HOERNER, 1965; SCHLICHTING, 1968)) indicates that the slightly lower Numbers attainable in the tunnel should not invalidate the findings as a model of full-scale conditions.

.~

0.7

~

0.6 0.5 Q,4

~ o~ 0 L

~

i _~ 0 2

~._u

>~ O' I 4 Diameter/height ratio,

5

dlH

Fig.3. Variation of the unobstructed roughness element drag coefficient (C~o) with the element d/H ratio. Where values o f the drag coefficient differ from that at a Reynolds' Number of 3.0 - 104, they are indicated by a single extending line (Re -- 2.25 • 104) or a double extending line (Re = 1.5' 104).

Methods Totalshearing stress (r). In the first method this was calculated from the momentum integral expression, where: h

r = I

h

u(u. 0

uldz

-

{I

~x

- (U/Uh) } dz

(6)

0

(TOWNSEND, 1956, p.263). Values of T were obtained for the unobstructed testsection floor and for roughness arrays with element spacings 1 < D / H < 5. The readings were corrected to a standard Uh of 20.3 m/sec assuming linear dependence of u(z) upon u,. For this correction a sensitive cup anemometer (cup centres at z = 28 cm) was used as the tunnel velocity monitoring instrument. It

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DRAG MEASUREMENTS IN ROUGHNESS ARRAYS

277

was positioned at x = 170 cm and 31 cm on the opposite side of the test-floor centre line from the Pitot-static tube and drag measuring positions (see below). Extremes of variation in tunnel velocity measured by the anemometer over several hundred runs were + 2.6 and - 1 . 6 %, with the majority of runs within the limits + 2 and - 0.5 % of the standard velocity. Velocity profiles were measured using a Pitot-static tube, from the freestream height through the turbulent boundary layer to within 0.5 cm of the test-section floor. The profiles were sampled at distances of 13.5, 53.3, 93.2, 133 and 180cm from the leading edge of the roughness field on a line displaced by y = 7.75 cm from the centre line of the test floor. Over the range of element spacing studied, averaging downwind along a single line provided similar results irrespective of the position of the sampling line cross-tunnel with respect to rows of elements. This was also SCHLICHTING'S (1936) experience. A second method was used for arrays of more widely spaced, regularly distributed elements and for all arrays of randomly distributed elements. Here, the total shearing stress was obtained by direct measurement, substituting a drag base of area F, 146.7 x 109 cm for the original test-section floor. The drag base was suspended by four 7 x 1.9 x 0.025 cm phosphor bronze flexings mounted 24 cm from each corner of the base so that its surface was level with the surrounding test-section floor. Its performance was evaluated by comparison with the m o m e n t u m integral method. The drag force on the base, W 1, was measured by applying a horizontal restoring force to return the base to its initial position in still air as indicated by a needle which magnified the base movement by approximately 30:1. The overall shearing stress, r, was then calculated using a corrected drag force, IV, as follows:

(7)

z = W/F

where: W=WI-cl

-

c2

(8)

with: cl = 69.35 (P1 -- P2) dynes and: c2 = N1 (~, ( N 1 ) -- wr) dynes The first correction, cl, allows for the static pressure difference between the upwind and downwind ends of the drag base, (PI + P o ) / 2 - (P2 + Po)/2, operating on the base end area 109.2 x 1.27 cm 2, where Po is atmospheric pressure. The second correction, c2, discounts the exaggerated drag caused by the first one or two rows of elements on the base. These were not included in the momentum integral evaluation, the first profile being sampled at x = 13.5 cm. N1 is the Agr. MeteoroL, 8 (1971) 269-292

278

J.K. MARSHALL

I

I t~ e.,

E 0

c-,

,q

,..o

,,.!.

Z

~8

0

t,tzl o

Z

0

.E ~3

.<

~

Z ev~ ..4

d

dddd

N

N

m

dd

~

N~N~

Z

~..~

~E ~

5

~

~

~J o I-

o~

z z

~3

c~

© e~

~ N d o d d d

d

oo

~E

Z

,-1 < >

Agr. Meteorol., 8 (1971) 269-292

DRAG MEASUREMENTSIN ROUGHNESS ARRAYS

279

n u m b e r o f elements upstream of this point, ~r (N1) is the mean drag force on these elements and V~, is the mean drag force on the remaining elements in the array. Both drag forces were determined by the method described in the following section. Typical values o f Ca vary from < 1 ~o to about 6 9~, being proportionally greater in the absence o f roughness elements; c2 falls off rapidly from about 2 0 ~ to a negligible proportion o f W1 with increasing element spacing, and is tabulated in Table III. Stress p e r unit floor area attributable to roughness elements. This was obtained by measuring directly the force on individual roughness elements, wr when:

(9)

W, IF = m, IA

The device used to measure the drag force on an individual element is shown in Fig.4. The thread passes over the pulley, and is attached to a weight which rests 2 Clll

P DP

L":..2t.Zl ' SA

a

--

P

2

Fig.4. Element drag measuring device. A. Plan, view. DP = drag pulley conveying thread to weight beneath floor; E = roughness element; P = jewel bearing pivot; SA = support arm; L1 = distance from pivot to thread attachment point; L2 = distance from pivot to point on support arm below the centre of gravity of the element. B. Section through a-b. CB = weight counterbalancing roughness element weight; F = wind tunnel floor. on the pan of a top-loading, sensitive balance. In still air there is no tension in the thread. When the wind tunnel is operating the drag force on the element is transmitted to the balance via the thread tension, to reduce the balance reading. This drag force: w, = (Mo -- M 1 ) 9 L1/L2

(10)

where ( M o - M I ) is the weight change recorded on the balance; L 1 / L 2 is the arm ratio o f the drag device (Fig.4A). D r a g on individual roughness elements was measured at several points Agr. Meteorol., 8 (1971) 269-292

280

J.K. MARSHALL

downwind of the test-section leading edge on a line offset by 16.5 cm from the profile sampling line. Measurements were made at nominal velocities of 10, 15 and 20 m/sec. Only elements in regular arrays were examined in this way. These results are presented in drag coefficient form using eq. 11, in Table III. For the calculation of W,/F, the mean of w, from 13.5 _< x < 183 cm was inserted in eq.9. The mean element drag force, ~, for x < 13.5 cm was used in the calculation of cz in eq.8. Roughness element drag coefficients, C I, were calculated using the measured drag force, w, when:

Cf = 2w,lpu 2 L~

(11)

for z = h, 3.94 H, 2.36 H and 1.6 H. Expression of the results in terms of several reference heights may enable application to the field situation, where there exists no convenient upper boundary condition analogous to the freestream velocity of wind tunnel flow.

Drag partition. The wall surface shearing stress, rg, was obtained by eq.6 and 7 for the unobstructed tunnel floor when: T ----1TO

Otherwise, Tg was obtained from eq.2: rg = (~ -

W,!t:)Fl~"

RESULTS

Total shearing stress obtained by different methods Fig.5 indicates satisfactory behaviour of the drag base over the range where

4

o m

f E~

~o

o,,/YoO 95 x

/

|

,,6"

~"~ ;oF

0

/

J

0.5 o

tunnel x

f~oor

10 20 30 40 50 Shearin 9 s t r e s s by dr~g base ,'ro(dynes.cm "2)

Fig.5. Comparison of total shearing stress obtained by the momentum integral method (TM) and by the drag base method (30).

Agr, Meteorol., 8 (1971) 269-292

281

DRAG MEASUREMENTS IN ROUGHNESS ARRAYS TABLE IV

COMPARISON OF TOTAL SHEARING STRESS:, "f, WITH THE STRESS PER UNIT AREA ATTRIBUTABLE TO REGULARLY DISTRIBUTED ROUGHNESS ELEMENTS,

VV'r/f AND

THE SHEARING STRESS AT THE INTER-

VENING WALL SURFACE, "fg FOR Uh = 2 0 . 3 M/SEC

Element diameter/ height ratio, d/H

Element spacing ratio, D/H

Unobstructed surface 0.5 b 49.5 1 2 hs c 2 3 5 0.5 1 2 hs 2 3 5 0.5 0.5 0.5 0.5 1 2 hs 2 3 3 5 0.5 0.5 2 hs 3 1 2 3 0.5 1 1 2 hs 2 2 3 3 3 5 5

49 48 48 47 45 34.5 34 33 33 32 30 24.5 20 14 10.5 13 12 12 14 11 15 6.5 5.5 6 8 6 6 6 2.5 5 2 1 4 1 5 3 1 5 2

Lateral cover (Lc • 10a)

0.2

7.40 4- 0.12 a 7.35

0.4 0.63 0.8 1.2 2.0 0.41 0.82 1.28 1.63 2.45 4.1 0.8 1.19 2.37 4.1 5.1 8.0 4.18 10.4 15.3 12.5 10.2 20.0 24.5 24.8 20.4 31.2 37.0 55.5 27.8 111 174 13.8 222 46.9 61.2 187.5 50 102

Total shearing stress, T (dynes/cm 2)

Roughness element drag per unit area, Wr/F '(dynes/cm ~)

Intervening wall surface shearing stress, Ta (dynes/cm2)

0 0.24

7.40 7.11

7.42 7.43 7.66 7.58 7.76 7.58 7.89 7.50 8.03 8.49 8.81 8.86 8.69 10.79 12.96 13.20 13.35 16.12 15.07 18.19 14.67 19.54 25.46 e 21.49 23.03

0.56 0.48 0.79 0.91 1.83 0.60 1.14 0.98 1.62 1.86 3.74 1.18 1.76 3.50 6.11 7.10 6.04 9.76 7.87 11.60 11.58 15.10 20.20 18.80 18.80

6.86 6.96 6.88 6.69 5.98 6.98 6.75 6.54 6.43 6.67 5.15 7.68 6.93 7.30 6.86 6.12 7.43 6.46 7.38 6.84 3.25 4.46 5.29 2.83 4.49

28.81 25.28 28.15 39.75 30.22 e 42.86 51.17 31.36 38.00 32.02 43.18 79.76 37.50 74.95

28.40 25.00 28.08 39.83 30.75 45.35 49.15 31.97 37.73 28.55 n.d. a n.d. 37.01 n.d.

0.41 0.29 0.08 --0.08 --0.54 --2.73 3.10 --0.67 0.41 3.90

0.61

Mean and standard error of twelve samples; b number without letters following refers to cylindrical element; e number with hs following refers to hemispherical element; a not reliably determined; e z450, d/HO.5, D/H 5.5, 23.7; d/H 1, D/H 5, 30.0 dynes/era2.

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J.K. MARSHALL

comparison with the momentum integral method was possible and consequently the remainder of the drag base results are considered reliable.

Partition of the total shearing stress The values at a velocity Uh = 20.3 m/sec, of the measured ~, Wr/F and rg, and the calculated zg for arrays of regularly distributed elements at different spacings and lateral covers are shown in Table IV. For the unobstructed tunnel floor, Wr = 0 and r = % = 7.40 dynes/cm 2 for Uh = 20.3 m/sec. As roughness elements are added, the total shearing stress is made up of contributions from both the wall and the roughness elements. Finally, at an element spacing ratio D / H of about 5-6, the condition z ~ W # F is reached. The data in Table IV are divided into these three categories by dotted lines.

Variation of the mean drag coefficient, C s (u~) of the roughness elements in arrays of regularly distributed elements When the regular roughness arrays are composed of sparsely distributed elements, the mean element drag coefficient for any one element shape remains practically constant with lateral cover, because the elements react largely independently of each other with the air movement. However, this unobstructed drag coefficient, Cyo, varies for the different elements (Fig.6). This variation is shown in relation to element d/H ratio in Fig.3 which also includes data at lower

0.7F

r

o/H o

,E3

E3

o.6 i

Q

0.5 1

¢ o.5 i

tA

2 • 2.hs J

0.3 g

ii

}o. i }

J

°o

,

2o

SO. RT reciprocal latera~ ccver

3~

4o

50

I/VrL-c

Fig.6. Variation of the mean roughness element drag coefficient, Cy(u,) with lateral cover as 1/~L,. Regression equation for region in which CI changes with Lc: Ci(un) = 0.0985/1/Le-0.134.

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DRAG MEASUREMENTSIN ROUGHNESSARRAYS

283

Reynolds' N u m b e r (see Fig.3 caption). No significant variation of Cy o with Reynolds' N u m b e r is apparent, and the variation with element d / H ratio conforms with the tendency demonstrated by TmLMANN (1953). The lower value of the drag coefficient for the hemispherical elements is also to be expected (TmLMANN, 1953; HOERNER, 1965). A sharp decrease in the value of C f takes place as the element spacing decreases. Here, the elements interact with each other through their effect on air movement. In this region of dependence of C j, on L~, the fall in C f is linearly, related to 1/x/Lc, i.e., C s = b/x/L~ + a. The constants, b and a, take the following values corresponding to the reference velocity heights h, 3.94H, 2.36H and 1.6H: 0.0985, - 0.134; 0.0989, - 0.12; 0.1014, -0.062; and 0.0988, 0.022 respectively. The transition in the behaviour of C f occurs at different values of 1/x/L c for the different elements. It extends over a finite range, but for practical purposes may be approximated by the point of intersection of the horizontal line C s = Cio with the sloping line C f = b/x/Lc + a, as demonstrated in Fig.6. Drag partition The value of zg is known when z = zg and the form of the partition of total shearing stress with changing element spacing follows from eq.2 when % and W , / F are both finite. The point at which zg ~ 0 is well illustrated by a plot of 1-~/(W,/zF) against loglo (1/Lc), a method of expressing the results suggested by Dr. R. A. Wooding. The data plotted in this way in Fig.7 show the condition zg ~ 0 to vary depending on element form and d / H ratio. Values for the constants, bl and al, in the equation 1 - x / ( W , / z F ) = bl logxo (1/Lc) + al when zg is finite, are given for the various element d / H ratios in Table V. Also presented in Table V are the values of Lc corresponding to the intersection points in Fig.6 and to the

TABLE V VALUES OF LATERAL COVER AT WHICH C f BEGINS TO DECREASE AND AT WHICH Tg BECOMES NEGLIGIBLE

Diameter/ Unobstructed Lateral cover Sample no. height ratio, drag coeff., when in partition d/H Cyo Cf = Cfos regression (Le" 10a)

Partition regression constants b bl

al

(Lc" 10a)

0.5 1 2 3 5 2 hs

0.423 0.397 0.421 0.431 0.504 0.422

--0.692 --0.618 --0.624 --0.595 --0.848 --0.590

23.1 27.8 33.1 41.7 20.9 39.9

0.64 0.60 0.43 0.33 0.39 0.34

16.3 18.0 30.8 45.8 34.6 43.2

7 5 4 6 3 4

Lateral cover when Tg --* O

a Obtained by substitution into CI (un) = 0.0985/1/Lc -0.134; b regression constants in regression of form 1 --~(W,/xF) = bx logxo(1/Lc) + al. Agr. Meteorol., 8 (1971) 269-292

284

J.K.

MARSHALL

0.9 ¸ d/H 0 D

0.5 l

• •

3 5

o.~

0.7

•

2 hs

0.£

I'

0

0.5p o

"

i

0.4P

._~

~

•

o

0,3 L

[3 L

-.~ o , 2 ~

"E

/z oll i:

-ojL o

1

_

2

I

I

3

4

Log reciprocal lateral cover lOglo 1/L¢ Fig.7. Variation of stress partition, as I--V'(Wr/TF), with lateral cover, as Ioglo (1/Lc). F o r regression equation constants see Table V.

points at which r o ~ 0 in Fig.7. Of significance is the general correspondence between these values of Lc, particularly considering the experimental data available and the large range of possible values of Lc. The general correspondence is of practical usefulness for two reasons. First, it allows an examination of other studies employing arrays of regularly distributed elements where partition of forces has not been measured, but only the trend of C I and Lc followed (see later). Secondly, on the basis of the assumption of general correspondence, "critical" parameter values can be defined at and beyond which the condition zg ~ 0 will approximately hold, and these can also be expressed in terms of the mean unobstructed drag coefficient, CIo, and lateral cover, Lc. The values of various roughness array parameters at this critical point are presented in Table VI taking as a basis for calculation whichever is the higher critical value of Lc in Table V.

Variation of the total shearing stress with element distribution The effect on the total shearing stress of aligning arrays of regularly dis-

Agr. Meteorol.,

8 (1971) 269-292

285

DRAG MEASUREMENTS IN ROUGHNESS ARRAYS

+

H

0 Z

.~ . ~ . ~ 0

•

Z 0

> 0 .<

~ rf I~I ,...q

.I

<

°

Z

.1 0

~-,,~ t ' l e,'~ ~e'. ¢',1

Agr. Meteorol., 8 (1971) 269-292

286

J.K. MARSHALL

tributed elements (d/H = 0.5 and 1, D/H = 5.5 and 5 respectively) at 45 ° to the direction of flow was examined using both eq.6 and the drag base. For neither element shape was the overall shearing stress substantially altered (Table IV, footnote). The results of total shearing stress measurements for arrays of randomly distributed elements and the corresponding regular arrays with the same element density are presented in Fig.8.

30

J

o

/

*

d

/A

~

o3

0.5

,)/

e

ii

cite

2Ps •

o

OIL 0

_ ~ fO

SheaPing stress

i ..... 20

o

~ 30

of random

~ _ 40

_ __j

50

aPrays

T r o n ( d y n e s . c m "2 )

Fig.8. Comparison of the total shearing stress for regularly distributed roughness elements.

('t'reg.)

and randomly (rra,.)

There is a tendency for the total shearing stress of random arrays to be slightly lower than for the equivalent regular arrays when the d/H ratio is 0.5,1 (Fig.8). As the element d/H ratios increase, so the correspondence becomes closer. The discrepancy found at the lower element d/H ratios may be due to the aggregating effect of randomising the elements (Fig.2). Such aggregation leads to a higher average d/H ratio for the array when the aggregations are treated as single elements. Consequent upon this, the expected reduction in C I for the randomised arrays of elements of d/H ratios 0.5 and 1 (Fig.3)could account for the difference. Aggregation of elements of d/H /> 2 would be expected to have less effect on the average element drag coefficient. For elements of any d/H ratio, aggregation will also result in a reduction in lateral cover. Consequently, arrays of randomly distributed elements may be expected to exert less drag than equivalent arrays of regularly distributed elements, but the results indicate that the difference is small (Fig.8). This finding can be examined in terms of the extremes within which regular and random distributions occur. The distribution exerting near maximum drag is one of a single row of elements aligned normal to the air flow, while near minimum drag is exerted by Agr. Meteorol., 8 (1971) 269-292

287

DRAG MEASUREMENTS IN ROUGHNESS ARRAYS

their alignment parallel to the airflow. A comparison of these extremes with regular and random arrays of cylinders was made using the drag base, and is shown in Table VII. Conditions were d/H = 2, twenty elements at a mean spacing ratio, D/H = 9. The difference between the random and regular arrays of Table VII indicates the effect, for a small number of elements, of chance alignment relative to the wind direction. However, the data in Fig.8 suggest that for practical purposes the results for regular arrays may be applied to random situations. This is important in the application of this work to natural vegetation. TABLE VII RELATIVE DRAG FORCE DEPENDING ON ELEMENT DISTRIBUTION IN RELATION TO WIND DIRECTION

Element distribution

Relative drag force

none

parallel

random

regular

normal

1.00

1.35

2.94

3.22

4.60

Comparison of results with those from other studies There are no other studies to my knowledge in which the force on the roughness elements has been measured independently of the total shearing stress. However, two studies have been made on regularly distributed roughness elements over a wide range of lateral cover. In a field study on the ice of Lake Mendota, KUTZBACH (1961) varied the lateral cover of 30 cm and 60 cm high roughness elements (bushel baskets) over the range 2 • 10-a to 290 • 10-3. He obtained the total shearing stress over these arrays by wind profile measurements under near neutral conditions at fetches varying from 18 to 80 m downwind of the leading edge of the array. His fetch to element height ratio consequently varied from 267 to 43 in comparison with 52 (at x = 133 cm) in the present study. However, his method of obtaining total shearing stress was more demanding on fetch requirements than the methods used here. His wind profiles indicated that he obtained internal boundary layers of thickness varying from about 160 cm to 180 cm. Kutzbach's wind profiles and friction velocity permit Co to be calculated with reference to heights up to z ~ 6H, and L c can be obtained from the geometry of the situation. For situations where zg is assumed to be negligible (Lc >/ 0.03) eq.5 can be expected to hold, and hence C: calculated. The results are compared with the wind tunnel results in Fig.9. Kutzbach's fourth run on 24 Feb. 1961 (A = 0.8 m 2) gives an estimate of C: markedly out of line with the other points and has not been included. The remaining runs show a degree of variability common in field data but nevertheless Agr. Meteorol., 8 (1971) 269-292

J.K. MARSHALL

288

0.7F 0.6!

't

O

& 0.5 J

i

0

ZV

X

/i'a

~'

/

8 o.3! L -o

rz[3/

I

oi5 o

x

• / •

3

*

7

O~

,/?

~J Kutzboch (1961)

T v 3.1)8 ~/Schlichting (1936) 2.13 O)

V

0~---

•

2Lt

/

o.,i w

d/H

/~"

.

.

.

.

.

.

.

.

.

.

l

2 3 4 5 SQ. RT reciprocol Ioterol ccver 1 /~/~c

.

6

7

Fig.9. Variation of roughness element drag coefficients, Cz (u2.36,) with lateral cover as l/]/Lc, from the present study and from data recalculated from SCHL~CHTINO(1936) and KUTZBACH(1961). Regression equation of combined data: Cs"(u2.86 H) = 0.0918/[/Le--0.0459. correspond quite well in trend with the wind tunnel results. His third run on 15 Feb. 1961 (A = 7.7 m 2) has been used to provide an estimate o f the unobstructed drag coefficient o f a single bushel basket (see Table VIII). SCHLICHTING'S (1936) study was carried out in a water tunnel 4 cm high with roughness elements varying in height from 21 to 41 mm. Typical values of Re = u(2mH/v are comparable with those in the present study, falling in the range 0.56 to 1.58 • 10 4. Schlichting calculated the roughness element drag coefficients over the range o f Lc(his F,/F) 7.85 • 10-3 to 907 • 10-3, referring his coefficients to the velocity at roughness element height. The calculation o f C I followed his eq.19 which, using the symbols o f the present paper, was: C s (un) = 2W,/Npu~L~ = 2(a/un) 2 [ ( u , / a ) 2 - Ft(u,,o/a)2/F]/Lc or: C s (un) = (~/Un) 2 (Co - FICg/F)/Lc

(12)

where ~ is the mean velocity o f the profile at the roughened wall. The results o f the present study indicate that the second term on the R.H.S. o f eq. 12 may be ignored for values o f L c > 0.03. Schliehting's element drag coefficients have therefore been recalculated in this range using eq. 12 in the form: C s (un) = 2(~/un)Zl.(u,/~)2/t c

(13) Agr. Meteorol., 8 (1971) 269-292

289

DRAG MEASUREMENTS IN ROUGHNESS ARRAYS TABLE VIII APPROXIMATE CRITICAL VALUES OF LATERAL COVER FOR A VARIETY OF ELEMENTS

Form

Roughness element diameter~height ratio, d/ H

height, H (cm)

Approximate unobstructed drag coefficient,

Approximate critical lateral cover g, Lc (crit.) • 10a

C1o(U~.3e~) Cylinder 0.5 Double bushel basketa 0.7 Cylinder 1 Single bushel basketb 1.4 Dense shrub e 1.52 Cylinder 2 Cylinder 3 Cylinder 5 Sphered 1 Hemisphere 2 Hemispherical segment e 3.08 Cone r 2.13

2.54 60 2.54 30 41 2.54 2.54 2.54 0.41, 0.21 2.54 0.26 0.375

0.66 ? 0.64 0.61 0.61 0.48 0.51 0.42 0.48 0.68 0.44

16.9 17.9 19.6 19.6 30.5 27.3 38.8 30.5 16.0 35.7

0.34 0.39

56.6 44.4

Drag coefficient taken as intermediate between cylinders used in this study; b KUTZBACH'S (1961) third run on 15 Feb. (A = 7.7 m 2) taken as approximate unobstructed condition without a significant contribution from zg; c dense shrub, Kochia aphylla from MARSHALL(1970) by applying eq.5 with CD (U2.a~H) = 0.0168; d from SCHLICm~NG (1936), plate XII, accepting Schlichting's use of eq.12 and recalculating Cs (UH) = 0.908 from velocity profile data to bring reference to 2.36 H; e from SCHUCHTING (1936), plate XIV, recalculated as above, Cf (UH) = 0.469; r from SCHLICHTING(1936), plate XXIV, recalculated as above, Cs (uH) = 0.561; g using combination equation C~ (uz.aeH) = 0.0918/1/Lo --0.0459. F r o m the profile a n d roughness element d a t a p r o v i d e d by Schlichting in his tables I, III, IV a n d V, values o f the new C f referred to velocities at o t h e r heights within the b o u n d a r y layer have been calculated a n d those at 2.36 H are included in Fig.9. Schlichting's results for spheres, hemispherical segments a n d cones c o m p a r e f a v o u r a b l y with the results from this study. R o u g h n e s s elements o f m a r k e d l y different width a n d depth, however, as with Schlichting's s h o r t a n d long angles, do n o t fall within the b o u n d a r i e s o f the established relationship o f changing C f with 1/x/L c. T h e results f r o m the present study, a n d those o f SCHLICHTING (1936) a n d KUTZBACH (1961) were c o m b i n e d in an equation o f the form: C f (Uxn) = b/x/Lc + a The values for b a n d a for velocity reference heights, 1.6 H, 2.36 H a n d 3.94 H are: 0.1014, - 0.0005; 0.0918, - 0.0459 a n d 0.0799, - 0.0656 respectively. A p p r o x i m a t e critical values for lateral cover for the elements included in the c o m b i n e d e q u a t i o n are presented in T a b l e V I I I as is the critical lateral cover for the shrub, Kochia aphylla, f r o m the study by MARSHALL (1970). Agr. Meteorol., 8 (1971) 269-292

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J.K. MARSHALL

DISCUSSION

The results presented are discussed in terms of their relevance to the problem which stimulated the study. This was to gain a better understanding of the susceptibility of partly vegetated surfaces to soil erosion by wind. A more analytical treatment of this and other work is provided in WOODING et al. (1971). Critical values A concept of critical values of various roughness array, and array element parameters emerges from this study. These are values at which the average surface force in the prevailing wind direction becomes negligible, a condition which is referred to as the critical condition. Definition of the condition requires a parameter describing the roughness array, such as lateral cover, and one characterising the roughness elements, such as their unobstructed drag coefficient. The behaviour of other element forms including tapering cylinders (bushel baskets), spheres, hemispherical segments and cones suggests that the concept of critical values applies to them also. If generally true for elements of similar width and depth, the concept has obvious implications for the problem of soil erosion from partly vegetated surfaces. Thus an area of ground several times larger than the average area occupied per vegetation element does not have to be completely covered for the average surface shearing stress to be negligible and hence, it is suggested, for soil movement out of the area to be arrested. Also, the concept indicates those properties of roughened surfaces to be taken into account in assessing site susceptibility to wind erosion in terms of vegetation cover. The form, size and shape of elements composing an array are shown to be important in deriving critical values (Table VIII). A qualitative picture of the efficiency of different elements in reducing the wind velocity emerges from Table VIII in which slender elements with a prominent upper edge are clearly superior to broad rounded elements. (The value for the drag coefficient of a sphere in Table VIII is probably high.) This picture should be helpful in generally assessing the efficiency of naturally occurring roughness forms. The efficiency of cylinders of varying diameter/height ratio has already been used to indicate a compromise solution to a specific range management problem (MARSHALL, 1970). The study provides what appear to be satisfactory experimental methods and analytical techniques for determining critical values. Arrays of elements arranged in a regular pattern allow the drag force on individual roughness elements to be measured conveniently. The demonstration that the condition of negligible T0 corresponds generally to the change in behaviour of Cf with Lc provides a useful technique for exploring the critical condition independently of measurements of total shearing stress. The method of plotting the results in Fig.7 to detect critical values of Lc appears to be extremely useful.

Agr. Meteorol., 8 (1971) 269-292

DRAG MEASUREMENTS IN ROUGHNESS ARRAYS

291

Non-regular arrays In naturally occurring arrays, the roughness elements are usually distributed in a non-regular, random or clumped manner. Interpretation of the behaviour of these arrays in relation to the better understood and experimentally more tractable regular arrays is of some importance. This study shows that, for practical purposes, it may be assumed that the behaviour of the average element drag coefficient and average surface shearing stress of non-regular arrays is similar to that of regular arrays of equal element density, correction being made, if necessary, when the element diameter/height ratio is less than 2. Field application of results Discussion of the field application of the results is appropriate because although initial field studies (KuTzBACI-I, 1961; MARSHALL,1970) compare favourably with the present findings some cautions are necessary. These include extension of the results to arrays composed of plants which show obvious changes in form, frontal area and permeability with increasing wind velocity, thus implying changes in Cy and Lc. A common field problem is the definition of a suitable height for element drag coefficient reference above naturally occurring arrays of elements of varying height. Part of the problem is deciding which elements to include when measuring array lateral cover. No arrays of elements of varying height were used in the wind tunnel study. However, a study was made of the contributions of equal numbers of cylinders of different diameter/height ratios to the total shearing stress. It was found that when the lateral cover of the smaller elements was 1/10 or less that of the remainder, their contribution to the total shearing stress was negligible. On this basis, shrubs of Kochia aphylla less than 20 cm high were neglected in the study of lateral cover and wind profiles over a surface roughened by them (MARS~IALL, 1970). Also in that study, the height of the roughness array was taken as the average of the maximum height of the shrubs contributing to the lateral cover. Having thus defined the roughness array height, the most useful range of velocity reference heights would appear to be 2 H to 4 H. Velocities at lower heights are more prone to distortion by flow patterns associated with individual elements. Velocities at greater heights, on the other hand, may fall outside the boundary layer characteristic of the roughness array and will also be liable to greater modification under conditions of non-neutral stability. ACKNOWLEDGEMENTS

The assistance of many members of the C.S.I.R.O. Division of Meteorological Physics, Aspendale, is gratefully acknowledged as is the use of that Division's wind tunnel facilities. Mr. E. L. Deacon provided valuable advice in the initial stages and, in his absence, Mr. R. J. Taylor provided continuing encouragement

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t h r o u g h o u t the experimental p r o g r a m m e . Mr. J. Stevenson a n d Mr. K. G r e a v e s assisted in the collection a n d processing o f the experimental data. The drag base was designed by the late Mr. C. J. S u m n e r a n d constructed in the Meteorological Physics Divisional W o r k s h o p . I a m particularly indebted to Drs. E. F. Bradley a n d R. A. W o o d i n g o f the C.S.I.R.O. Division o f Plant I n d u s t r y for e n c o u r a g i n g me to extract m o r e i n f o r m a t i o n from the experimental data, for stimulating discussions a n d for critical a p p r a i s a l o f the manuscript. REFERENCES ANDERSON,D. J., 1967. Studies on structure in plant communities, 5. Pattern in Atriplex vesicaria communities in southeastern Australia. Australian J. Botany, 15: 451-458. BAGNOLD,R. A., 1941. The Physics of Blown Sand and Desert Dunes. Methuen, London, 265 pp. CHEPIL, W. S. and WOODRUFF,N. P., 1963. The physics of wind erosion and its control. Advan. Agron., 15: 211-302. C.S.I.R.O. (AUSTRALIA),1967. Division of Meteorological Physics, Annual Report (1966-67). C.S.I.R.O., Melbourne, 36 pp. GREIG-SMITH, P., 1964. Quantitative Plant Ecology. Butterworths, London, 2nd ed., 256 pp. HOERNER, S. F., 1965. Fluid-Dynamic Drag. S. F. Hoerner, Midland Park, New Jersey, N.J., U.S.A., 452 pp. KUTZaACH, J. E., 1961. Investigations of the modification of wind profiles by artificially controlled surface roughness. Wisconsin Univ.. Dept. Meteorol., Ann. Rept., 1961: 71-113. MARSHALL,J. K., 1970. Assessing the protective role of shrub-dominated rangeland vegetation against soil erosion by wind. Intern. Grassland Congr., l l th, Surfers Paradise, (Australia), 11: 19-23. SCHLICHTING, H., 1936. Experimentelle Untersuchungen zum Rauhigkeitsproblem. Ingr. Arch., 7(1): 1-34. (Translation: N.A.C.A. Tech. Mem., 823). SCHLICHTING,H., 1968. Boundary Layer Theory. McGraw-Hill, London and New York, N.Y., 6th ed., 747 pp. TILLMANN,W., 1953. Neue Widerstandsmessungen an Oberflachenst6rungen in der turbulenten Grenzschicht. Forschungsh. Schiffstech. 1: 81-88. TOWNSEND, A. A., 1956. The Structure of Turbulent Shear Flow. Cambridge Univ. Press, Cambridge, 315 pp. WOODING, R. A., MARSHALL,J. K. and BRADLEY, E. F., 1971. Drag due to regular arrays of roughness elements of varying geometry (in preparation).

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