the influence of gravity and eventually reaches a terminal velocity. ... erties of the fluid and the sphere ï¼the terminal velocity and the terminal distance can be ...
Movements of a sphere rolling down an inclined plane Mouvements d'une sphere roulant sur un plan incli的 CHYAN-DENG lAN ，Associate Pr 可essor ， Departmelll of HydrauLics and Ocean Engineering ， NationaL Cheng Kung Universi 妙，Taiwan. JINN-CHYI CHEN ，PH .D. Stude 叫 D 叩artment of Hydraulics and Ocean EIψneeri 峙，的 tional Cheng Kung Universi 妙，Taiwan.
ABSTRACT Released from rest on an inclined smooth plane in a stationary fluid ， a sphere accelerates along the plane under the influence of gravity and eventually reaches a terminal velocity. The variations of velocity with time and distance ，the terminal velocity ， the terminal distance (the practical distance required for a sphere from rest to its terminal velocity) ， are investigated through laboratory experiments and a theoretical analysis. The relationship of the drag coefficient and the Reynolds number for the moving sphere with its terminal velocity is obtained and compared with that in the free fal l. The effect of proximity of sidewalls of the flume on the fluid drag acting on the steady movement of the sphere is evaluated. The terminal velocity and the terminal distance against the sediment number are presented in dimensionless graphs. Given bed inclination as well as the properties of the fluid and the sphere ，the terminal velocity and the terminal distance can be determined directly from the graphs. The experiments of the steady movement for a sphere rolling down a rough inclined boundary are also presented.
RESUME Llichee a partir du repos sur un plan incline lisse dans un fluide egalement au repos ， une sph 色re accelere sou I'influence de la pesanteur et atteint eventuellement une vitesse limite. Les variations de vitesse en fonction du temps et de la dist 也lce parcouru 巴， la vitesse limite ，la distance limite (c'est-a-dire la distanc 巴， mesuree a p 位tir du repos ， necessaire a I' obtention de la vitesse limite) ont ete etudiees au moven d'essais en laboratoire et d'analyses theoriques. Laliaison en 甘e coefficient de trainee et nombre de Reynolds de la sphere en mouvement en fonction de sa vitesse limite a ete obtenue et comparee a ceUe de la sphere en chute libre. L'influenc 巴 de la proximite des parois laterales du canal sur la trainee de la sph 色re en mouvement permanent a ete evaluee. La vitess e4 limite et la distance limite sont foumies en fonction d'un parametre adimensionne l. A partir de la don nee de la pente du lit ainsi que des caracteristiques de fluide et de la sph 色悶， la vitesse limjte et la distance limite peuvent e 甘e determinees directement a partir de graphiques. L'article presente enfin des essais en mouvemenl permanent pour une sph 色re descendant par roulement Ie long d'un plan incline rugueux
1
Introduction
An immersed
body experiences
fluid drag when it moves
fluid ， or when it rests in a flowing form drag. The former
fluid. The drag consists
is due to the friction
of the body's
di 仔'erence across
the body. It is generally
On 巴 is interested
in the total drag and rarely desires
surface
drag except
difficult
for some sediment-transport
FD is usually expressed
at a relative of a surface
to separate
CD
to its surrounding and a
and the latter to the pressure
the fom1 drag and the surface
to compute
investigators
in terms of a drag coefficient
surface
velocity
drag (or skin friction)
the form drag separately
drag.
from the
(lan and Shen ， 1995). The fluid drag
which needs to be empirically
determined.
'，‘、
) l
AU
呵，‘
FJ
J﹄
OF
V A
D
$﹄司司
C
D
一一
F
Revision received May 12， 1997. Open for discussion till April 30 ， 1998
IQUES. VOL. 35. 1997. NO.5
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in which PI is the fluid density; Ao is the area of the particle projected onto a plane nonnal to V and equalsπD2/4 for a sphere of diameter D; V is the relative velocity between the particle and its surrounding fluid. Theoretical analyses of the drag coefficient for a sphere steadily and slowly falling through an infinite quiescent fluid had been done by some investigators (Graf ， 1971， Chapter 4). However ， theoretical drag coefficients are available only for very low Reynolds numbers. For higher Reynolds numbers ， there is no analytical solution for the drag coefficient ， and thus various empi 吋cal and emi-empirical equations of the drag coefficient have been proposed by previous investigators. In general ，the drag coefficient for a body moving in a fluid depends on various factors ，such as the Reynolds number ， the shape of the body ， the proximity of the boundary ， the concentration of suspended particles ， and the intensity of fluidωrbulence (Garde and R;mga Raju ，1985). A considerable amount of infonnation is available concerning the variation of Co for a body moving in a fluid with the abovementioned factors. However ， there is little infonnation about the variation of the drag coefficient for a body moving in contact with a boundary or in the vicinity of a boundary ， such as a sand pa 此 icle moving along the sea bed or the bed of an alluvial channel (bed load transport) ，a vehicle used for undersea exploration ， a rock moving in a debris flow ， among others. Carty (1957) started the study on CD for a sphere rolling down an inclined smooth boundary with a terminal velocity in a quiescent fluid ， such as water and oi l. His experiments with water were conducted in a 0.76 m wide ， 0.91 m deep ，1.83 m long rectangular flume and the experiments with oil were conducted in 0 .4 6 m wide ，0.1 m deep ，0 .4 6 m long flume. Spheres used in his experiment were lucite ， glass ， steel ， and cellulose acetate spheres. These spheres varied in diameter from 2.0 mm to 31.6 mm and in specific gravity from 1.17 to 7.82. The flume width W was much larger than the sphere diameter D ， and the ratio DIW ranged from 0.003 to 0.055. CD was calculated by using Eq. 12 that will appear in the section after the next section. Carty's results show that CD for a sphere rolling down a smooth boundary is much larger than that in the free fal l. Similar experiments were conducted by Garde and Sethuraman (1969) in a 0.23 m wide ， 0.60 m deep ， 3.50 m long flume. Spheres used in their expe 吋 ments were glass ， s
2 Equation of Motion fOI Consider that a sphere of d quiescent fluid. The sphere the plane is 6. Under the ac， eventually reach a terminal sphere include fluid drag f tance FR from the inclined mass In， rolling down an in tion of the forces parallely 1
、~
J::
Q正
Fig.
I.
Schemat. ic diagramof and rough boundarie
WBsin6 - FD-l
or
句
J
D
nb
O
rJ nur
。、 、
、
nv
-s'
， f π-rb
The first term on the left haJ sphere along the plane. Th e taneous velocity V. Th e thir
690
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， Vo L. 35.1997.NO.5
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plane normal to V and the particle and its surldily and slowly falling 1971， Chapter 4). HowImbers.For higher Reyis various empirical and evious investigators. In 'actors， such as the Rey centration of suspended A considerable amount I a fluid with the abovehe drag coe 仟icient for a as a sand particle movvehicle used for under1
mooth boundary with a with water were conIe experiments with oil ed in his experiments d in diameter from 2.0 V was much larger than {ascalculated by using ow that CD for a sphere ilar experiments were 旬， 3 .5 0 m long flume. If ith diameters varying ~erthan the one used in :h is paper，as was done ion of their experimenabout two times larger ly not be known ， Garde times longer than that :l use it permits both the motion of the sphere. Ie and Sethuraman wa ~of DIW would have a ect would result in an tio DIW and the flume e a discrepancy in the aper is aimed to evaluI terminal velocity on a :h e drag coefficien t. In is also evaluated.
Ll QUES. VO L. 35，1997. NO.5
2 Equation
of Motion for an Accelerating
Sphere on a Smooth Plane
Consider that a sphere of diameter D initially at rest on an inclined smooth plane is rei巴ased in a quiescent fluid. The sphere density is p" while the fluid density is PI' and p" > Pi' The slope angle of the plane is 8. Under the action of gravity ， the sphere will b巴gin rolling down the inclined plane and eventually reach a terminal veloci 吟， if the inclined plane is long enough. The forces acting on the sphere include fluid drag FD' fluid lift FL' submerged weight of the sphere" 勻， and rolling resistance FR from the inclined plane ， as indicated in Fig. I. Performing a force balance on a sphere of mass m.， rolling down an incline from rest ， we applied Newton's Second Law to obtain ， by projection of the forces parallely to the incline plane
(a) on smooth boundary
Fig. I.
(b) on rough boundary Schematic diagram of the forces acting on a sphere in a quiescent fluid rolling down inclined smooth and rough boundaries.
The first term on the left hand side of the above equation is the submerged-weight component of the sphere along the plane. The second term is the fluid drag for steady motion ， evaluated at an instantaneous velocity V.The third term is the inertia force of added mass that is included because accel-
JOURNAL OF HYDRAULIC
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eration of th 巴 sphere involves acceleration of th 巴 fluid. t is the time and CA is the added mass coe 征 icien t. The fourth term is the rolling resistance from the boundary which can be obtained through the analysis of the balance of mom 巴n t. It is noted that this term is absent in the case of a spbere in th 巴 free fall since there is no rolling resistance from boundary. The left hand side of the equation represents the forces acting on the sphere and the right hand side is the mass of the sphere times its acceleration. Rearranging the above equation gives
a + bCoV·
dV
=互J
(4)
in which
Using the chain rule dVldl =
V;+I = V;+(al
where Xi + 1 and Xi means the tance L (i.已， travel distance evaluated by
2: (X;+I-J
L =
，
a=g(ps- p )sin8. 一 l .4 p.， + CAP
b
=-3ρf
， '
(5)
4D( 1. 4ps + CAP，)
The terminal distance Lis al flume length is less than 出 reaches to the end of the flu
For the case of a sphere in the free fall ， the coefficients a and b are replaced by
α=g(Ps-Pf): Ps + CAP/
b =-3p
3 Terminal Velocity for a
，
(6)
4D(ps + CAP，)
3
'。
rJ
O
ny
(7)
一九)
D 。 、‘''，
);(/;+1
ns v
2
V;+I = V;+(a+bCoV
( π-6
Wh en using Eq. 4， reference must be made 伽 flo叮r a graph of 伽 d ra嗯 gc∞ O 巴叮描仔扭ICαIe 圳 n叫t Co vs Rey卯 y吋吶怕 no 昀 01比 d由sn叫Il山 u叩 1汀 I汀 m m伽 b Th 旭巴 e Reynolds number is defined 的 a s Re=VDI內 v. Hence ， CD is simply dependent on Vas the sphere diameter D and 伽 fluid viscosity v are specified. Th e nonlinear term in the Eq. 4 precludes fmding a closed form solution because CD varies with V. For the numerical solution ， dVldl is rewritten as (V i + 1 - Vi)/ (Ii + 1 - Ii) where the subscript i means 出e step of time ， and the equation of the motion becomes
When a sphere rolls down a ThEequation ofrrlotion ofti
Rearranging Eq. (J I) for the
en
~包 -PI]
=
U3pfvf Use of Eq. 7 in solving an accelerating sphere problem requires a stepwise solution that involves finding drag coefficient CD from 出巴巴 mpirical relationship of CD and 丸， and added mass coefficient as wel l. In each small time interval (t i + 1 - tD ， CD is explicitly evaluat 巴d with Vi' Th eoretically ，both the total elapsing time and travel distance needed for a sphere from rest to its terminal velocity are infinite. If the sphere velocity Vi at time step i = M reaches 99% of the terminal velocity 吭，出 e sphere is practically considered to be in the movement with terminal velocity. The elapsing time T (named terminal time here) for a sphere from rest to its practical terminal velocity can be evaluated by M
T=
692
芝 (t;+I-t;)
(8)
JOURNAL DE RECHERCHES
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L. 35，1997，NO.5
for the terminal velocity V， ~
時(
V ，=
The above equation as it 吼叫 V， also and thus the value G method). However ， rearrang
JOURNAL OF HYDRAULIC RESEAl
\d CA is 出 e added mass r which can be obtained is absent in the case of a he left hand side of the is the mass of the sphere
(4)
Using the chain rule dV/dt = (dV/dx)( dx/ dt) = V(dV/dx) ， the discreted equation of motion is
1
Vi+ I
= Vi + (a V-+ bCD
V)i(Xi+
I -
Xi)
(9)
where Xi + I and Xi means the location of the sphere at time Ii + I and fi ， respectively. The terminal distance L (i.e. ， travel distance required for a sphere from rest to its practical terminal velocity) can be evaluated by
(10)
L =玄(九 I - x ，)
(5) The terminal distance L is an important information for the design of the experimental flume. If the flume length is less than the travel distance ，由e sphere can not reach a terminal velocity before it reaches to the end of the flume.
~dby
3 Terminal
(6)
vs Reynolds number Re' t on Vas the sphere diam，Precludes finding a closed is rewritten as (只+I-V;)/ ~motion becomes
Velocity for a Sphere Rolling on a Smooth Plane
When a sphere rolls down a smooth incline at its terminal velocity 吭，the forces are in equilibrium. The equation of motion of the steady movement of the sphere is
D
:(psff)gD3si
件
Cn
=
~(Ps 一ρl)gD~
AULIQUES ， Vo L. 35. 1997. NO.5
0
(I I)
，~
( 12)
4ps- P/gD h: 3 PI CD- 一-
( 13)
D-5pfvfaEH
(8)
吶 D2V~ =
Rearranging Eq. (11) for the drag coefficient CD gives
(7)
olution that involves findadded mass coefficient as 們.Th eoretical 旬，both the terminal velocity are infivel∞i句吭， the sphere i apsing time T (named terbe evaluated by
:C
for the terminal velocity V， give
V ，=
“自
The above equation as it stands is not convenient for practical applications because CD depends on V， also and thus the value of V， must be determined by successive approximation (trial and error method). However ， rearrangement of Eq. (13) gives
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VO L. 35. 1997. NO.5
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2e
R D C
N
司、 一-
andf
自
fc. Therefore the eq
given by ( 14) '‘
s
OO
d
nv-nv
D
)，'， nv
'，
寸
nv ''，‘、 πτb
d-AU
in which the dimensionless parameter N equals [(Ps _ pJ)gD3sinS/(pJv2)] and is named as the "sed iment number" here. Note that Co is a function of R. ， defined as Re = V ， D Iv. If the relationship of CD and Re is known ，one can develop the relation of Nand Rt. Knowing 恥，D ，PJ' v and S (and thus N) one can determine from the N - Ree relation the of R. and thus the value of V ， without a trial . -.-~....... _..- value . _._and 巴rror method.
or
...，" '"1'
vv
Fu
，
tEE''
If a sphere rolls down an inclined rough boundary ，的 indicated in Fig. Ib ，there is resistance to the movement of the sphere from the boundary due to the collision and sliding friction between the sphere and the boundary. Assuming a sphere steadily rolling down a rough (bumpy) boundary made of identical spheres and analyzing the balance of energy of the sphere's movement ，Ian and Shen (199S) obtained the expression of the resistance due to the collision and the sliding friction.
、、 D
of a Sphere on a Rough Incline
丹、以-AUT
SEESt
、
FJ
+
4 Steady Movement
For a sphere rolling down Eq. 17 is reduced into
V ，= JgD(Sinl F
口
=:fJsD2vf+μ(W8
( ISa)
FL)
叫一
where fe is a coefficient relating to the collision while μis a coefficient relating to the sliding friction. The lift force FL can be expressed as FL =π/SCLp jJ 2V ，2and the Ufe coefficient CL to be determined. Rearrangement of Eg. (ISa) yields that the resistance consists of two parts: one is velocity dependent and the other velocity independen t. The velocity-dependent part is proportional to the squares of the velocity and the velocity-independent part to the submerged weight component normal to the boundary.
nu
nu 、 p 。
c
o
w
+
u﹒
9-g
v
s
nv
D
2
FJ π?0
s
=
Fι
(ISb)
where fis a coefficient relating to the collision and lift force ，f = fc 一(3 仲']C d4p ，). Based on the datafc 0.3S to 0 .4 9 (Fig.9) ，μ 0.001 to 0.07 (Fig. I0) ，CL O.IS (Aksoy ， 1973) and 0< PJ/ p，< 0.63 in present experiments ，one can see thath. is much larger than (3 仲'~d4p ，).刊 i implies that the effect of lift force on the resistance can be disregarded in the present experiment 國
國
圖
for tanS>μand V， = 0 if I~ such as air ，is negligible ，th dent of th巴 sphere density roughness that is representl rewrite Eq. IS into
Y = AX + B
where Y= V.尺gDcosS) 斗， ~ data (X ， η，we can evaluate friction coefficients throu determined ，CD for a sphe~ obtained by the following 1
Cn D
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= 4~gD(p ， 一一 -
JOURNAL OF HYDRAULIC RESEJ
andl !C. Therefore the equation of motion for a sphere steadily rolling down a rough boundary is given by 揖
( 14)
aJ
'，‘、
、o ‘ r
nu
LV
--
尸
no 。 3
3
。
D
rJ
OO
AV
、，
s
HF
0.
'，.‘、
呵，&
π-8
2r
S
OO
V D
C
nv
FJ
21
FJ
π-6
勻，&
V D
D
J
or
:c仙一(lf)(
(t e+ 芳 :h ere is resistance to the ng friction between the ugh (bumpy) boundary 舟 's movement ， Jan and Id 出 e sliding friction.
nv
nb
C π-8
可
J
no n
rJ
OO
D
)
s
n﹁
nv
(
π-6
I is named as the “ sediIf the relationship of CD PI' v and 8 (and thus N) lue of V， without a trial
叫一μcos8)gD
=
0
( 17)
For a sphere rolling down an incline in the air in which P.. is much larger than PI' i.e. ，pip.. « I， Eq. 17 is reduced into
h
JgD(
沁 μcos8)
( 18)
(I Sa)
obESE-t
n-Die san d OMn on
cvιμm
It--13
巾i!
.叫LO叭叭
htmoo 1rc
nHρ-v-t-guv
etRUns-
-UO?mAIV
0.m 戶pu g 仇附.則﹒叫
Mcd 川必
EI B --: 叫1
UVρLVρ
、
戶
for tanS>μand V， = 0 if I tan 81 :Sμ. Eq. 18 implies that when the influence of the interstitial fluid ， such as a 甘， is negligible ，the terminal velocity of a sph 巴re rolling down a rough incline is independent of the sphere density but depends on the sphere's diameter ， th巴 bed inclination ， and the bed roughness that is represented by Ie and μ. In order to evaluate .t;. and μusing such experimen 吟， we rewrite Eq. 18 into
Y
( 15b)
~/4ps)· 0.15 (Aksoy ， 1973) and than (3μpFd4p ，.). This the present experiments
=
AX + B
where Y= 吭 2(gDcos8) 斗， X = tan8 ，A =Ie-I and B =一 μtfc-I. According to a series of expe 討 mental data (X ， η，we can evaluate A and B by a linear regression analysis and obtain the collision and the friction coe 仟icients through the relations Ie = IIA and μ= -BIA. Once Ie and μhave been determined ， CD for a sphere rolling down a rough incline in a fluid (such as water or oil) can be obtained by the following relation
Cn = ~gD(ps o 一尋
lJ LJQUES. VO L. 35. 1997. NO.5
( 19)
JOURNAL OF HYDRAULIC
- P/)(sin8
一μcos8)~P.f
PI昕
RESEARCH. VOL. 35. 1997. NO.5
f
;JC
一豆豆
(20)
695
5 Laboratory
Experiments
Experiments of a solid sphere of diameter D rolling down an incline were performed in a rectangular flume with a smooth bottom ， as shown in Fig. 2， to study the effect of the proximity of the sidewalls of the flume on the drag coefficier 此， the transient motion of a sphere from rest to its terminal velocity ， and the relation between the drag coefficient and the Reynolds numbe r. Experiments were also conducted in the rectangular flume with rough bottom made of a layer of identical spheres of diameter K closely packed to each other in order to evaluate the influence of the relative bed roughness D/K on the drag coefficien t. 什
180 em
..
viscosity relation for the sala a controlled temperatu陀 wa( of pure water at 20°C. Si阿 increase of temperature. Ext rate for salad oil， indicating I In experiments ， a sphere ~ allowed to roll down as a r( video-camera recorders at ( one to determine the transI! 0.033 seconds. Spheres use
6 Experimental
Results
6.1 The influence of proxi
A billiard ball of diameter influence of proximity of i.e. ，W = 5.70 cm ，6.20 cm D/W varied from 0.22 to O. nal velocities of the sphere widths were measured. Wi sphere as well as the bed Reynolds number R.(= VI I: both with R. and D/W. FOl ligible. On the other hand ， of D/W. The value of CD a the sidewall effects on 出巴 and the relation of CD and
(a) on smooth boundary 180 em
oj
16
Fig. 2.
(b) on rough boundary Schematic diagram of a rectangular flume used in studying the movements of a sphere rolling down inclined smooth and rough boundaries
12 Q
Th e flume with transparent sidewalls was 0.40 meters high and 1.80 meters long ， and tiltable from zero
ω15 degrees. Th e flume could be a句 ustedωany desired width between zero and 24.8 cm. Transparent 5 mm x 5 mm square grid papers were placed on the 甘ansparent sidewalls of the flume so that the exact position of any moving object in the flume could be 甘aced for observation and measuremen t. During experiments ， the flume was filled with a Newtonian fluid (either pure water or salad oil). Properties of the fluid to be determined were its density and kinematic viscosity ， both of which may vary with temperature. Water used in experiments was tapped pure wate r. Th e temperature-viscosity and tempera 仙陀density relations of pure water are available in most standard textbooks of fluid mechanics. Salad oil used in the experiments was a commercial produc t. Th e speci 日c gravity of salad oil was measured by a Welch Specific-Gravity Balance in a controlled temperature water bath. Th e density of the salad oil was then obtained from the measured specific gravity multiplied by the density of wate r.甘 le density of the a1 ad oil is 0.892 glcm3 at 20.5°C ， and slightly decreases with incl它的e in tempera 仙reo A temperaωre-
696
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L. 35，1997.NO.5
U ...r Z U
U
也 g
.
m
υ
∞
。 咽 L 咽
JOURNAL OF HYDRAULIC RESE
erformed in a rectanguIeproximity of the sidefrom rest to its terminal nber. Experiments were of identical spheres of the relative bed rough-
viscosity relation for the salad oil was determined by using a digital viscometer of an accuracy of ± I% in a controlled temperaωre water bath. Th e kinematic viscosity of salad oil is about 65 times more than that of pu 陀 water at 20°C. Similar to water ，the viscosity of salad oil decreases exponentially with the increase of temperature. Experiments also show that the shear stress is linearly proportional to the shear rate for salad oil， indicating that the salad oil is a Newtonian fluid in the room temperature [Jan， 1992]. In experiments ， a sphere was laid on the bed of the upstream end of the inclined flume and then allowed to roll down as a result of its own weigh t. The motion of the sphere was recorded by 8-mm video 也amera recorders at differ 巴nt locations along the flume. Time cod 巴s on video cassettes enable one to determine the translation velocity of the moving sphere. The time codes have an accuracy of 0.033 seconds. Spheres used in this study were billiard ball ， steel spheres and golf balls.
4
6 Experimental
Results
6.1 The it呎 uence of proximity of sidewalls on Co
呀
-“
A billiard ball of diameter of 5 .4 7 cm was used as a rolling sphere in experiments to investigate the innuence of proximity of sidewalls of the nume on Co. Seven values of flume width W were used ， i.e. ，W = 5.70 cm ，6.20 cm ，6.80 cm ，9.10 cm ，12.4 0 cm ，18.60 cm and 24.8 cm ，and thus the ratio D/Wvaried from 0.22 to 0.96. The bed inclination in experiments varied from 2° to 10°. The teml 卜 nal velocities of the sphere rolling down the inclined flume at various bed inclinations and flume widths were measured. With the measured terminal velocities and the properties of the fluid and the sphere as well as the bed inclination ，the drag coefficients CD were determined by Eq. 12. The Reynolds number R.(= VP/v) varied from 50 to 20 ， 000 in this study. Figure 3 shows that Co varie both with R. and D/W. For D/Wless than 0 .4， the influence of the variation of D.乃 Von CD is negligible. On the other hand ，for D，八¥ larger than 0 .4 0，CD monotonously increases with the increase of D /W. The value of CD at D /W= 0.96 is about twice larger than that at D /W= 0.22. For avoiding the sidewall effects on the movement of the sphere ， when evaluating the added mass coefficient CA and the relation of CD and R. ， the experiments have to be conducted with D /W n ce the values of D ，恥，
lnd pip! = 2.65 ，and the I to 1. 29 X 107. With the en the terminal distance . On the other hand ，for 16.The corresponding G ~ for the sphere 叫 lin 8L/v ，which is much Imum terminal distance 1 (that is the result of a which is about half the from rest on the incline
data ，D ，Sand 吭， the relations between Y = V，2(g DcosS)-1 and X = tanS at various inverse of relative bed roughness (D/K = 1.10 to 4.15) are indicated in Fig. 8. Th rough a linear regression analysis ，one can obtain the coefficients A and B of Eq. 19 and thus the values of !c and μ. It is found that both!.. and μdecrease with the increase of D/K ，的indicated in Figs. 9 and 10. Onc 巴!c.. and μhave been determined ， CD for a sphere rolling down a rough incline in a fluid (such as water or oil) can be obtained from Eq. 20. Figure 11 shows that CD for a sphere rolling down a rough incline is close to that down a smooth incline at low Reynolds number (R~ < 200) but the former is about twice larger than 出巴 latter at high Reynolds number (見>2000). Figure 11 also shows that the CD curve obtained from Garde and Sethuraman (1969) is quite lower than that obtained here. The discrepancy comes from in their empirical model for the resistance Fes to the movement of the sphere from the rough boundary. In their model ，Fu is expressed as Fes = [O.4 (D I.的 .5 -O -C .0610gRe]WLJCosS. There are two main dificiencies in their model for Fe.' One is that 丸， is negative if O.4 (D f，的-C.5 < 0.0610gR ，. It is unreasonable for Fe. having negative values. A further dificiency in their model for Fe ，.is the fact that it does not account the collision e 仟'ect on Fe.，.Shown in Eq. (15a or 15b)，the model for Fe "， proposed in the present paper includes the e 仟凹的 of collision ，friction as well as life forc 巴，and h巴nce the result obtained here should be more reliable than that from Garde and Sethuraman.
0
LLYMU
instantaneous translation velocity of a sphere at time r = Ii; terminal velocity of a sphere; width of the experimental flume; submerged weight of a sphere; distance along the inclined plane; dimensionless parameters relating to Eq. 19; P. PI sphere density and fluid density; v fluid kinematic viscosity; inclined angle of a smooth or a rough boundary; μfriction coefficient relating to Fe.，.; Vi
V， W WB x x，Y
B. T. SMITH ， Engineerill~ R. ETf EMA ，Iowa I f/ sri 52242 ， USA
SUMMARY Laser-Doppler-velocimelry i and roughness influence ove mappings show lhat skin frio initially ，in response to covel ate mechanism whereby an ~ t巾 utions of velocity and Rel (which suggests ice-covered covered flow is not accurate RESUME Des mesures au velocimelre couverture de glace monlrent et la structure de I' ecoulemi montrent que la rugosile de reponse a la couverture de gl. elre un mecanisme par lequel egalement que les dunes mod pial jusqu'a un niveau tel que ne soit plus adaptee a des c~ courverture de glace peuvent
1 lntroduction
Temporal-mean structure: 陀searchers ， notably Raudk (1993). Th e pu 中ose of th tructure over two-dime rJi approach taken entailed USI sion aJtrain of dunes under ing of the study is provide presence on bedform mo 巾 LDV m 臼 surements is sim experiments documented b reflects an equilibrium for presented do not reflect eq open-water flow over dune respons 臼 in bedform dime' Revision received June 12，I
