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The motion of cobbles in the swash zone on an impermeable slope
COASTAL ENGINEERING Coastal Engineering
33 (1998) 41-60
The motion of cobbles in the swash zone on an impermeable slope P.A. Luccio a, S .I. Voropayev a,1,H.J.S. Fernando a, D.L. Boyer a,*, W.N. Houston b aEnvironmental Fluid Dynamics Program, Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 852874106, USA b Environmental Fluid Dynamics Program, Department of Ciuil and Environmental Engineering, Arizona State lJuiuersity; Tempe, AZ K5287-6106F USA Received 27 May 1997; revised 20 November
1997; accepted 26 December
1997
Abstract The purpose of this communication is to present the results of a series of laboratory experiments aimed at better understanding the dynamics of the motion of large bottom particles (cobbles) in a swash zone. In this region, a thin sheet of water that results from the collapse of a turbulent bore, runs up the beach and can induce the transport of relatively large solid objects in the on-shore direction. The aims of the study were to: (i) mimic this process in laboratory experiments and identify the associated physical processes invoived; and (ii) to deveiop a suitabie theoretical model to describe the motion of cobbles. The experiments employed a solid impermeable bottom and were conducted in a long tank of rectangular cross-section. An impulsive hydraulic bore, produced by a dam-break mechanism at one end of the tank, was used to simulate the water motion in the swash zone. Solid objects of simple discoid shape were used to model the cobbles. The results of the laboratory observations were compared with model predictions. In the range of external parameters used for the experiments (size and density of cobbles, propagation velocity and height of the water front, slope and friction at the bottom), a reasonable agreement between the measured and calculated values of the cobble displacement as a function of time was obtained. 0 1998 Elsevier Science B.V. Keyw0rd.r: Sediment transport;
Cobble dynamics
* Corresponding author. Tel.: + l-602-965-1382; fax: + I-602-965-1384; e-mail: [email protected]. ’ Also at the Institute of Oceanology, Russian Academy of Sciences, Moscow, 117851, Russian Federation. 0378-3839/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PIZ SO378-3839(98)00003-9
42
P.A. Luccio et al./ Coastal Engineering 33 (1998) 41-60
1. Introduction Sediment transport in the near-shore zone is of great importance for many engineering and geophysical applications. During the past several decades, extensive research has been conducted on the study of the nature and dynamics of surface waves and their interaction with the ocean floor (e.g., see Longuet-Higgins and Stewart, 1964; Engelund and Fredsoe, 1982; Peregrina, 1983; Craik, 1985; Battjes, 1988). The resulting knowledge has led to the development of sophisticated models of the wave-induced motion fields and accompanying transport of fine sediments under wave action (e.g., see Yalin, 1977; Mei, 1982; Mei and Liu, 1993; Sleath, 1984; Fredsije and Deigaard, 1992; Neilsen, 1992; Belorgey et al., 1993; Arcilla et al., 1994). Nonetheless, significantly less attention has been paid to the study of the motion of much larger objects, such as cobbles. The American Geophysical Union (e.g., Sleath, 1984) has classified cobbles as sediment particles with typical diameter D = 6.4-25.6 cm. Apart from obvious engineering and geophysical applications, the transport of such large particles as cobbles is a for anti-tank mines in the topic of current naval interest; i.e., devising countermeasures near-shore zone (e.g., see Lott and Poeckert, 1996). The near-shore zone may be divided into three subzones, wherein different mechanisms dominate the dynamics of the transport of bottom particles (e.g., see Battjes, 1988). As waves approach the beach parallel to the shore line, they induce oscillatory motions in the water column, thus producing unsteady forcing on particles resting on the bottom. These waves then either break or do not break. In the former case, the breaking waves generate a highly turbulent surf zone characterized by strong vortices and substantial vertical motion. The vortices can be sufficiently strong to detach rather large cobbles from the bottom. In the latter case, they collapse at the shore edge, without forming a surf zone. The breaking waves are characterized as being either plunging or spilling waves; further in shore, both types of waves eventually develop into a turbulent bore. As these hnrt=c nnnrnnrh the v”Lvu ..yy”-v-A .. . . wnt~rlim= ..I-v-l...l) thev “--, mllnnw v-*-“ry-
and ______ form IA I.. cwnch mn~ _-__ _Y__-----,
in yhi& __.
a &in s&et
of water resulting from the collapse runs up the sloping beach. In the swash zone, considered part of the surf zone, the beach is periodically covered by a relatively thin layer of water. The fluid motion here is strongly asymmetric: during the on-shore motion phase, the energy concentrates into a thin water sheet with a relatively sharp front which, when impinging on a cobble, may transmit significant amounts of momentum to the cobble, thus initiating its motion. The return phase, which occurs after the maximum wave run up, is controlled mainly by gravity and appears to be somewhat less energetic. Recognizing that in different regions of the near-shore zone, fundamentally different background motion fields determine the dynamics of the transport of large bottom particles, three separate experimental installations were constructed at Arizona State TT-:..___:r.. _&..,a..*L.. __1^_.^_& pal&K --~:,.1- &_ __^_^ _.&Lllt;LllillllSlllb. -,.,.L..-:“-‘p-,.,J-, “s~llla~“ly ,.“,.:ll..&--. “‘,,“t;lblLy 6,. L” bllllly LIK IeleVallL LlillqJ”IL I” III”Ur;l flow, a standing wave tank was constructed; to simulate the direct effect of breaking waves, a 33.5-m long tank in which progressive waves impinge on a sloping beach was developed; and to model the swash zone, as discussed herein, a dam-break tank was used. In the present paper, we report only on laboratory experiments obtained from the dam-break facility, which are akin to studies on cobble dynamics in swash zones.
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
43
The present work has been motivated by the desire to predict the motion of isolated large bottom particles (e.g., anti-tank mines) on smooth sloping beaches. It is noted that most natural cobbles in the oceanic environment are located in regions for which the overlying cobbles, which may be free to move, are resting on cobble beds. The motion of such (cobbles is a problem of interest but not the one to be addressed in the present communication. Furthermore, natural beaches are permeable with bedforms that change shape in the presence of currents and wave action. Nevertheless, owing to the relatively unexplored nature of research on the motion of large bottom particles in the oceanic environnnent, a reasonable starting point for research in this area is to consider an impermeable beach. Restricting to an impermeable bottom both simplifies the problem at hand imd, additionally, in due course, will aid in better understanding the dynamics of cobbles resting on a permeable sandy beach. In the present study, we restrict to cobble motions in the swash zone. Results of experiments with oscillatory flows akin to those conditions in the shoaling region (Voropayev et al., 19981, as well as for breaking waves, again with impermeable beaches, are to be reported separately. The remainder of the paper is organized as follows. The experimental installation and methods of study are described in Section 2. The results of the experimental observations are given in Section 3. A model is derived in Section 4 and comparisons with the experimental data are discussed in Section 5. Finally, the main conclusions are formulated in Section 6. 2. Experimental
installation and methods of study
As mentioned, a dam-break zone. In this approach, a dam of water, is instantaneously well-defined and reproducible rectangular cross-section. By h&&t onrl nrtxnom&nn cna=rl “U’~“C ULlLb yLvyY~uu”” 0yuAa
method was used to simulate the water motion in a swash or gate, which, prior to the experiment, blocks a reservoir or nearly instantaneously removed. This results in a bore which advects through the uniform channel of varying the height HO of the water behind the dam, the nf “I
the LllU hnre ““LU
r-an -cut
ho vu
~vGPA flZ*‘6. .c&LIUU \r
11 1,.
Two similarities can be noted between a breaking wave impinging on a beach and a turbulent bore; i.e., shape and speed. Apart from these large-scale visual similarities, there is also an internal dynamical similarity; namely, the energy lost through the dissipation in a bore gives realistic estimates of the energy lost in the surf zone (e.g., see Fredsije and Deigaard, 1992). These similarities are the principal reasons the dam-break method was chosen to model the water motion in a swash zone.
6)
Fig. 1. Schematic diagram of (a) the experimental -cobble, 4-gate, 5-water.
@I set-up and (b) model cobble; l-tank,
2-sloping
bottom, 3
44
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
The experiments were conducted in a rectangular Plexiglas tank having a length of 250 cm, a height 60 cm and a width 30 cm (Fig. 1). A vertical removable Plexiglas gate was installed at a distance of 60 cm from the end of the tank and this smaller section was tilled to the desired level (H,, = 5-30 cm) with slightly-dyed tap water of room temperature (density p = 1 g cme3, kinematic viscosity v= lo-’ cm* s-l>. Two separate tank floors were used in the experiments; i.e., smooth (Plexiglas) and rough, for which a thin layer of sand was glued to the Plexiglas. This permitted a variation of the kinetic friction coefficient K between the bottom and cobbie by a factor of four. The values of K (0.1 f 0.01 and 0.4 + 0.04) were determined directly by slowly dragging a model cobble in still water and using standard dynamometric methods to measure the resulting friction force. By tilting the bottom at a small angle cx to the horizontal, the effect of a sloping bottom was produced in the experiments. Model cobbles can be characterized by a diameter D, a mean height h, and density p,. The model cobbles of discoidal shape, with rounded comers of radii = 0.5 cm, D = 10 cm and h, = 4-6 cm, were hollow and were made using wooden forms; the materials used for the outer shell were a polyester resin and fiberglass. By filling the model cobbles with different materials (sand or polystyrene or glass spheres), it was possible to change the mean cobble density p, from 1.5-2.4 g cmp3. In the experiments, one of the cobbles was placed at a fixed distance (50 cm) from the gate in the empty part of the tank. To initiate an experiment, the gate was then rapidly removed vertically. The side view of the cobble motion was recorded using a super VHS video camera located at a large distance (4-5 m) from the tank; this assured that parallax errors would be minimized. In some cases, a standard 35-mm camera was also used. The video tape was then digitized frame by frame using commercially available digital processing software (DigImage: Dalziel, 1993) and all experimental data (positions of the cobble and height and positions of the water front) were obtained from these digitized images using standard data processing techniques. Each experiment lasted only l-2 s and, without such rapid digitizing procedures, the errors in measurements would Inroe rhp tvniml in -__ rnhhk __--__ lnrdnn ___- ________ ,._-__ have __I._ hem v ____ tnn ___ _-p” ‘, r __ I_ errnr - -- _ - in ___ &em_inip_v e the time and space from the measured values of the instantaneous cobble displacement x,(t) using DigImage was approximately 2-3%. A total of 41 experiments were conducted; the results of these experiments are discussed in Section 3.
3. Results of observations This section will be used to provide a discussion on the background present qualitative observations of bore/cobble interactions.
flow and to
3. I. Background flow Consider the background flow produced by the dam-break method. It is useful to note that the classical solution of this problem, derived by Bitter in 1892 (e.g., see Whitham,
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
b
0
Fig. 2. Slcetch of the dam-break flow. Solid line-water bottom friction at the tip of the front.
52
4.5
x+
surface without friction, dashed line-correction
for
19741, has no bore (Fig. 2). After the gate, which is at x = 0, is opened at t = 0, the water surface takes a parabolic shape with the height h (x, t) given by:
h=-f- 2( gH,)1’2 9g [
and the fluid particle velocity u=3
2r
(gH0)1/2+fl, L
1
x 2
- ;
(1)
u along the x-axis given by:
(2)
I
where g is the gravitational acceleration, H,, is the initial water height behind the gate I’* _< w/w < 2(gH,) ‘I2 . The front of the flow, at x = x+, has zero and -(g&J height (h, = 0) and propagates in the positive direction along the x-axis with the velocity u + equal to the fluid particle velocity u at this point:
dx+
u+= dt
(3)
= 2( gH,)“*.
The point where h = H, propagates
with the velocity:
. ._
dx_
‘-= - dt = -(
in the opposite direction
gffoy,
(4)
while: k!=%H,andu=c(gH,)l,2atx=O.
(5)
In a real flow (e.g., which will be depicted in Fig. 5a below), the height h, of the tip of the front is finite (Fig. 2) and propagates with the velocity u0 = dx,/dt, which is less than u+. These differences were explained by Whitham (1955), who considered frictional resistance near the tip of the front, resulting from turbulence, which dominates the flow dynamics in this region. His modified boundary layer solution shows that h,, as well as uO, remains approximately constant through the tip of the front and: /I, = a,H,,
(6)
where a2 is a slowly increasing function of time, with a2 = l/4 at small times. This prediction is in very good agreement with our measurements shown in Fig. 3. The values of h, in all experiments conducted at different values of H,, were taken at the
P.A. Luccio et al. /Coastal
46
Engineering 33 (1998) 41-60
Fig. 3. Height, ho, of the water front as a function of water height Ho all experiments were taken at a distance 20 cm behind the front, when from the gate. Symbols-experimental data: (0) smooth horizontal dashed lines-approximation (Eq. (6)) with a2 = 0.2 (smooth bottom) measurement errors are given by the vertical bars.
behind the gate. The the front was at the bottom, (0) sandy and a2 = 0.3 (sandy
values of ho for point x = 50 cm sloping bottom; bottom). Typical
moment when the front reached the point x = 50 cm and the height h, of the front was measured at the same distance (20 cm) behind the leading edge of the front. Although the coefficient a2 increases when the bottom friction increases, for the range of H, used in the experiments the results of measurements can be approximated by the simple dependence (Eq. (6)) with: a2
=
constant = i
0.2, smooth bottom, 0.3, sandy bottom,
and this parameterization is shown by the dashed lines in Fig. 3. Whitham (1955) also predicted that at small times, the frontal rapidly from u + = 2( gH, )I/* to the asymptotic value:
(7)
velocity
ua drops
where a, = 2/3 at intermediate times. This is also in agreement with the results of our measurements shown in Fig. 4. Using the data on the position of the front x,, as a function of time t, mean values of the frontal velocity IQ, = dx,/dt, were calculated for small times (at x 5: 50 cm) and are shown in normalized form in Fig. 4 for experiments __-XL._r_J a~ ..&UIIIE;ICIIL XCc^-n-c _._I ..^^ VI -C no. u -_ .I&_UI ^CIL__.. ^^^.__^-^-._ give -:..- L?,r 1.4 fCK CVII~~CL~U villues rut; _^^. IGSUILS ulest: -~~~t;asuro~ueu~s both smooth and sandy bottoms (see Fig. 4). From the general theory of internal bores arising in two-layer fluids, it follows that in the limiting case of an external bore (one fluid layer), a, and a2 are related by a: = (1 + a,)/2a, (see relation (4) in Klemp et al., 1997). This gives the estimates: a, = 1.7 for a2 = 0.2 and a, = 1.5 for a2 = 0.3, which are in reasonable.agreement with the measured values.
P.A. Luccio et al./ Coastal Engineering 33 (1998) 41-60
n *I 0
1.5 -
e
00
-c--_-------o~-o0
0.0
a,=1.4 o---,,
n
n
a I
d
l-
$
0.5 -
01
0
I
I
I
I
I
I
5
10
15
20
25
30
Ho(cm) Fig. 4. Non-dimensional front velocity, u0 /(gH,) ‘I2 at a small distance (X = 50 cm) from the gate for different values of If,,. Symbols-experimental data: (0) smooth bottom, (0) sandy bottom. Dashed line-approximation (Eq. (8)) with a, = 1.4. The values of u. = dxO /d t werecalculated using the experimental data on the front position x0 as a function of time; a typical error bar is shown.
The fluid particle velocity u behind the front is less than the frontal velocity larger than the fluid velocity at x = 0, thus:
u0 but
(9) Recognizing that a, drops rapidly to the asymptotic theoretical the following, the fluid velocity u is approximated by: 24 =
Ul( gfl,y,
value with a, = 2/3,
in
(10)
where ‘zl = constant = 2/3. 3.2. Motion of the cobbles In the experiments, the cobbles were placed at a fixed distance (n = 50 cm) from the gate along the tank axis (Fig. 5a). The gate was opened and the water front was created. The distance from the gate to the initial position of the cobble was the same (50 cm) in all _-___. mns. -_
Thin ___-I dictnnre - _-_____
wa< ..-- chnnen ___- -___ emnirirnllv ____r____-__,
in ~r&r
TV nmvide r--.---
R tin of qprgxi~_gte!y - -*r
constant height as the water front reached the cobble. Upon impingement of the water front (Fig. 5b), the cobble started to move (Fig. 5c) with a velocity U = dx,/dt, where X, is the cobble displacement from the initial position, x, = 0 (x = 50 cm) at t = 0. Typic&/ plots of the cobble displacement x, as a function of time t for runs with different initial conditions are shown for a smooth horizontal and a sandy sloping bottom in Figs. 6 and 7, respectively. In these data, the time origin t = 0 was chosen at the moment when the cobble started to move; the cobble displacement was measured from the initial position of the cobble x, = 0 at t I 0.
48
P.A. Luccio et al. / Coastal Engineering 33 (I 998) 41-60
Fig. 5. Sequence of photographs showing the motion of a cobble in a dam-break flow: (a) top view, (b-c) side view; (a) t < 0, (b) t = 0, (c) t = 0.04 s, (d) t = 0.67 s. The flow moves from right to left. The time origin, t = 0, is chosen at the moment when the cobble, which was initially at rest at x = 50 cm, starts to move. Experimental conditions: H, = 20 cm, D = 10 cm, h, = 4 cm, p, = 1.5 g cm13; smooth horizontal bottom.
As can be seen in these figures, the cobble starts to move with some finite initial velocity, U, # 0 at t = 0; i.e., the cobble motion is an impulsive one. This initial velocity is relatively large and depends on the initial conditions of the experiment, but remains significantly less than the mean velocity U of the water sheet. As a result, the
49
P.A. Luccio et al. /Coastal Engineering 33 (1998) 41-60
Fig. 6. Typical experimental data (symbols) showing the cobble displacement xc as a function of time t for a smooth he3rizontal bottom. The time origin t = 0 is chosen at the moment when the cobble, which was initially at X, = 0 (X = 50 cm), starts to move. Dashed lines-the model estimates (see below). Experimental conditions:(+)H~=18cm,D=10cm,h,=6cm,p,=2.0gcm~3;(~)H~=20cm,D=10cm,h,=5 cm, pC = 2.4 g cm ~3;(~)H~=20cm,D=10cm,h,=4cm,p,=1.7gcm~3;(O)H~=29cm,D=10 cm, h,=6cm, pC=1.7gcm-3.
water front passes the moving cobble and continues to move with approximately constant velocity ahead of the cobble (Fig. 5~). For the case of the smooth horizontal bottom, the cobble velocity U gradually increases and reaches an approximately steady
1.6
Fig. 7. Typical experimental data (symbols) showing the cobble displacement X, as a function of time t for sandy sloping bottom. Dashed lines-the model estimates (see below). Experimental conditions: (0) Ho = 20 cm, D = 10 cm, h, = 4 cm, pC = 1.5 g cmu3; (*)~,,=23cm,D=10cm,h,=4cm,pC=1.5gcm-3;(0) H,=29cm, D=lOcm, h,=4cm, p,=1.5gC11-~.
50
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
value at t = l-l 3 s (Fig. 6). For the sandy sloping bottom, U remains constant or decreases slightly with time (Fig. 7). At larger times (not shown in Figs. 6 and 71, because of the finite length of the tank, the water front reaches the end wall of the tank, reflects and starts to move in the opposite direction. This leads to a rapid decrease of the cobble velocity, and in some runs conducted at relatively large values of the water height H, behind the gate, a reverse motion of the cobble was observed. To eliminate the influence of the reflected motion, only the data when the front moved from the gate to the end wall were used and compared with the estimates predicted by the model. As can be seen in Figs. 6 and 7, the cobble displacement for the same time, say _x: at t = 1 s, depends strongly on the initial conditions and changes by a factor of four from _x: = 20 cm to .x: = 80 cm. The values of xi, as might be expected, increase for increasing H, and decreasing pc. One of the interesting observations is that the bottom frictional coefficient K does not influence significantly the initial cobble motion. This can be seen clearly in Fig. 8, where cobble displacements x, are shown for two cases for which all experimental conditions except for bottom friction were the same; i.e., K = 0.1 (smooth bottom) and K = 0.4 (sandy bottom). In spite of this relatively large change in the value of K for the two experiments, the resulting change in the cobble displacement with time is less than 20%; see, for example, t = 1 s, showing that bottom friction does not influence significantly the dynamics of the cobble. Detailed visual observations show that at relatively high propagation velocities, the cobbles may be detached, at least partly, from the bottom and this effect may significantly reduce the influence of bottom friction. This is not to say that the value of K does not influence significantly the initiation of cobble movement. Recall that both n, and t are zero until the cobble movement is initiated.
i.:
Fig. 8. Cobble displacement xc as a function of time t for experiments for which all parameters were the same, except the bottom friction coefftcient, K: (0) smooth bottom (K = 0. l), ( + ) sandy bottom (K = 0.4). Other parameters: Ho = 23 cm, D = 10 cm, h, = 4 cm, pc = 1.5 g cmm3, (Y= 0”. Dashed lines-model estimates: upper-for smooth bottom, lower-for sandy bottom.
P.A. Luccio et al,/ Coastal Engineering 33 (1998) 41-60
51
A total of 41 experiments were conducted with cobbles using different values of the external parameters Ha, p,, h,, D, a and K, where Ha is the water depth behind the gate, p,,, h, and D are the cobble density, height and diameter, (Y is the bottom slope and K is the kinetic bottom friction coefficient. In addition, 16 experiments were conducted without cobbles to obtain the data on the background flow characteristics (Figs. 3 and 4). In the experiments with cobbles, some threshold value of Ho must be exceeded to initiate the cobble motion (see the estimates below), and in these runs, H, was varied in the range H, = 9-29 cm. The values of the other parameters were set as follows: p, = 1.5, 1.7, 2.0 and 2.4 g cmm3, h, = 4, 5 and 6 cm, D = 10 cm, cx= 0 and 2” and K = 0.1 (smooth bottom) and 0.4 (sandy bottom). A model for cobble motions was developed concurrently with the experimental studies and is presented in Section 4. The proposed model can be used to calculate the cobble displacement X, as a function of time t and other external parameters. The results of these calculations are compared with the experimental data.
4. A model and some estimates The model to be described here was developed essentially from basic principals while incorporating certain parameterizations based on present and past empirical results. The system under consideration is shown schematically in Fig. 9. The cobble is initially (t I 0) at rest, with its center at X, = 0. A water sheet of depth h and density p propagates with velocity U along the slope and at the moment t = 0 impinges on the cobble. During the impact, the water transmits some momentum to the cobble and, if
Fig. 9. A schematic of the model flow configuration: (a) side view, (b) top view. Incoming water front propagates from left to right with velocity E(t) along the sloping bottom and at t = 0 impacts initially still cobble, which is at x, = 0. During the impact, the water transmits some momentum to the cobble which is initially at rest and at t > 0, the cobble moves with the velocity U(f) = dx, /dt, which changes with time, because of the bottom friction and positive (U < 7i) drag force.
52
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
this momentum exceeds a threshold value, the cobble starts to move with some velocity U,, in the positive direction of the x-axis, which is parallel to the bottom. Owing to bottom friction and the drag force, the propagation velocity of the cobble U = dx,/dt changes with time t. We wish to determine the dependence of the cobble displacement X, from its initial position on the time elapsed and other external parameters. To derive the governing equation, assume that the fluid everywhere in the neighborhood of the cobble would have had the velocity U in the absence of the cobble. This is the standard assumption used to describe
(11)
In deriving this expression, it was assumed that the acceleration reaction (Batchelor, 1967) on the cobble is equivalent to MK, d/dt(U - U), where M is the mass of water displaced by the cobble and K,( = 1) is the virtual mass coefficient. The total inertia M,(dU)/(d t) + MK,d/dt(lJ - U) is thus assumed to be balanced by the buoyancy contribution G sin (Y, where G = (M, - M)g, the pressure gradient contribution M(dE)/(dt), the total drag force on the cobble Fd and the bottom friction force Ff. It is expected that the pressure gradient force on the cobble is the same as that on an equivalent fluid mass, which, for the inviscid limit, is the same as M(dZ)/(dt). Denote p * = p,/p, A = ii - U,
(12) and use M = (rr)/(4h tions: 1 r”d = zps,(ii Ff=
* )ph, D2, M, = (~)/(4)p~
h, 0’
and the standard parameteriza-
- Uj2C&
I.“\ (‘3)
a-
(14)
-lK(Gcos
EF,);
here, S, = h, D/h * is the cobble cross-sectional area as it is immersed in the fluid, C, = C,(Re) is the drag coefficient, Re = (U - U)D/u is the instantaneous Reynolds number of the cobble, K is the kinetic friction coefficient at the bottom,
FL = ;pS,(ii
- U)‘C,
(16)
is the lifting force arising when h, < h,, h * = 1, C, = C,(Re) is the lifting coefficient, S, = TD~/~ and: z=
EF,,
1,Gcos
a>
0,Gcos
CYIEF~.
(17)
53
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
In derivmg Eq. (15) it was assumed that when h, > h,, the principal contribution to the lift force is the drop of pressure along the top of the cobble where the water velocity relative to the cobble exceeds the water velocity away from the cobble. When h, < h,, it was assumed that there is no water above the cobble and, as a result, no drop of pressure. This leads to the parameterization (Eq. (15)). Relation (15) assumes that when h * > 1, h, > h,, there is no lifting force, and Eq. (17) shows that the bottom friction Ff become;3 negligible when the lifting force exceeds the buoyant weight of the cobble. Eq. (11) can now be written in the form:
(18) To solve Eq. (18) it is necessary to specify the functions C,(Re), and C,_(Re) and the water motion which can be characterized by the mean velocity U and depth h, of the water sheet. Let the water sheet propagate along the sloping bottom as a slab of thirknw:c h. = mnctant~ renrewntd hv. ...A~~y.~l,”“” vV”“.-.r) the= .&._ velnritv . v’.,.?“J ran w-1 he -1 ‘“y’w”“““.. v-7. ii = u - gt sin ff ,
(1%
where Eqs. (6) (7) and (10) can be used for h, and u. In deriving Eq. (19) the action of the gravity force on the water sheet on the sloping bottom was taken into account. In the absence of the bottom, the drag coefficient C, for a round cylinder moving in a large volume of water with velocity A = S - U, with good accuracy can be approximated as C, = 1.0 for lo* I Re I 2 X lo5 (see, e.g., White, 1991). In the present experiments, this corresponds to the range: 0.1 cm s-l I A s 200 cm SK’, where C, may be considered as a constant (D = 10 cm, u= lo-* cm* s-’ were used for this estimat’e). In the presence of the bottom, the drag coefficient may change. For example, *l__ ?I___ ___~A-:_:__rc__ _ __L___ :________ L_. ^ C--r_.. _l? _---__:--r-l_. 1 c -I l- *l__ UK urag C”~lllCltx,L 101 a spert: IIIl.xcils~Y uy a la~lul WI appruxmkmxy I .J--L III UK presence of the bottom. But it can be shown (see, e.g., Eagleson and Dean, 1959) that such an increase of C, for a sphere can be attributed at least partly to the additional friction because of the rotation of a sphere rolling along a sloping bottom. We acknowledge that in general, the coefficients C, and C, are complicated functions of geometry and other external parameters, with typical values C,, C, = 0.5-l (see, e.g., Yalin, 1977). For the geometry of the present application (solid bottom, finite depth of fluid), exact values of C, and C, are unknown and for simplicity these coefficients were approximated as follows:
(20)
C, = c, = constant = 1 .O, which were used to solve Eq. (18j and to derive quantitative motions. For U given by Eqs. (19) and (18) for t > 0, becomes: dA itdt + BA* = C,
estimates
of cobble
(21)
54
P.A. Luccio et al./ Coastal Engineering 33 (1998) 41-60
where for brevity: A=(p*h*
+K,),
(22) c = ZgK( p * h * are lused .
Tha lllcl
- 1)cos CY,
anlnt;nn LI”lULl”ll
nf “I
E’n Yy.
i31i \tiI,
.wh;,-h nlllrll
cca~ the rnn&t;nn ilcaCI0IIUD Cl&cl ~“IIUIII”II
A z Y -
A Y()
.zt b t=~ UC
n “+,
P’J” tic&II
be easily found and, for example, for E, Z = 1: I
_ Cl/2
B’/2A,
1-t
e(-2B’/2C”Z/A)r B’/2A
0
B1/2A
1+
+
Cl/2
_
Cl/2
(23) (-2B”2C”2/A)t
Bl/~~
+
Cl/2
\
To calculate the initial value of A,, = U - U,,, we must estimate the velocity U,, of the cobble just after (t = 0,) the impact of the water front, which has at the moment of impact (t = 0) a velocity U = u given by Eqs. (10) and (19). To estimate U,, one may ~~_~ IL- r_-rl- __. .wltn ..CL1.r,.. .___--_1 f--.-.-_r _l? _ _-If> L-J_. ._.L1_L -_- _.^^ ._.:*I_ Al__ _._I__:*_. use me dndqy me nunndl impdu 01 a xuiu oouy, wmm IIIUVCS WIUI LIIGve~ocuy U, on a free surface of liquid. Suppose, that at t < O_ the water sheet is at rest and its front (vertical free surface in our case) is approximately at the center of the cobble (see Fig, 9b). At the moment t = O_, the water starts to move and during the short time interval, O_ I t 4 O,, its velocity increases from zero to u. In the ideal case, this process can be described by a singularity similar to the delta-function. For this short time interval, the leading term on the right side of Eq. (11) is the first term. Integrating Eq. (11) over the short time interval, O_ I t I O,, and neglecting the other small terms on the right hand side, we obtain an estimate for the initial velocity of the cobble:
u =p 0
(1+4n) (p*h*
-H&J
u,
where the coefficient /3 < 1 is introduced to take into account that the cobble starts to move into free space, where there is no liquid at the moment of impact. It can be shown (e.g., see Batchelor, 1967) that: p = constant = l/2 for bodies of simple geometry. 2
(25) ’
In deriving Eq. (24), several effects were neglected: i.e., (i) the splashing of water when the front impinges on the cobble, (ii) the presence of the solid bottom on the lower side of the flow and the free surface above, and (iii) bottom friction. It is not at all clear how to include all of these very complicated factors into the physicai modei; aitemativeiy, it is preferred to use the simpie estimate (Eq. i24)) to derive realistic vaiues for Us. In fact, the coefficient p in Eq. (24) cannot be derived analytically for the present case; neither can it be considered as a universal constant. The average value p = 0.5 was chosen empirically by comparison of the measured (using the slope dx, /dt at small t) and the estimated values of Ua. Only by chance did this value of /3 coincide with Batchelor’s estimate (Eq. (25)). Some justification for this simplification is the fact that at least in the range of parameters used in the present experiments, only one value for p had to be used (see below).
55
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
Thus, taking into account that the cobble velocity is:
we have for the cobble displacement:
(27)
x,=jnfIidf-/nlAdr,
where the first integral gives the position of the water front and the second gives the distance from the water front to the center of the cobble, which is behind the front for t > 0. Csing Eq. (19) for U and Eq. (23) for A, the integrals in Eq. (27) can be calculated and the cobble displacement x, can be presented in closed form: I @B
1 x,:=ut--ggt*sina+ 2
1/2C”2 /A)t
C II2 A Vn i B i
_
(BYA,
-
(BW,
+ P)
1 _ (B1’2do(B%,
\
~7~‘~)
P)
’
’
+ Cl’*) (28)
where A, B and C are determined by (24). Before comparing model estimates consider certain limiting case. Assume the cobble are negligible (i.e., C,, K = limit B, C + 0 in Eq. (28), one obtains
Eq. (22), A, = U - U,, and U,, is given by Eq. with the experimental data, it is instructive to that the viscous drag and the bottom friction on 0) from which B, C = 0 in Eq. (22). Taking the for the cobble displacement:
1 lim x, = U,t-Tgt2sinff.
(29)
k3,C-t0 ~XTI__-
WIltm
..-1..
0111y
Al__
l__n_-
Lilt: UULLUIIl
c:_r:__ IIICLIUII
_Al__ 011 UK
__l_l_l_
CUIJDlC
:_
-__,:_:l_,_
IS Ilegllglulc,
---
UIlt:
c-_a_. 1IIIuS;
1 lim x, =ui---gr2sina-~ln(l+~Aor). C+O
For the case when both the drag and lifting forces are unimportant: 1 C lim x, = U,t-Sgt’sincY-xt’coscu. B+O
(31)
At small times, Eqs. (30) and (31) coincide with Eq. (29) and the cobble moves with the initial velocity, U,, in all three cases. At larger times, two effects begin to play a role: (i) acceleration (the last term in Eq. (30), which must be expressed to an accuracy of rr/.?\\ 1_ -P *l .. . _~. ufr-~~: Decause or me posiuve viscous drag (U < u), and (ii) deceieration (the iast term in Eq. 1(31)), because of bottom friction. For typical values of the parameters used in the experiments, these terms are comparable at intermediate times, and one may expect that there will be some balance between these terms. Thus, the drag, as well as bottom friction (and hence, the lifting force) are important and neither of these effects can be neglecied in the complete solution (Eq. (28)).
56
P.A. Luccio et al./Coastal
Engineering 33 (1998) 41-60
It is clear from the physics that to initiate the motion of the cobble, the value of H, must exceed some threshold value, say H * . This threshold may be estimated as follows. To calculate U,, Eq. (11) was integrated over a small time interval, and the last two terms in Eq. (11) were neglected. This approximation is correct for relatively large values of H,,, when the first term in Eq. (11) exceeds the friction term. By estimating the time of impact as r = D/u and integrating Eq. (11) over this time interval, it is possible to obtain an approximate criterion for the initiation of motion (dU/dt > 0): P(l+K,)Mu=(M,--M)gr(sin(~+K,coscu),
(32)
and this gives: (huh*-l)D H*
z
(sincr+K,coscu), (1+
(33)
K,)M
K is the static friction coefficient where Eq. (10) was used for u and K, = (1.52), (when the cobble is not sliding). In deriving Eq. (33), it was assumed that during the impact Fd and FL, which enters into Ff in accordance with Eq. (14), can be neglected. For the typical range of parameters used in the experiments, Eq. (33) gives the .. cm for a sandy estimates: ii * = 2-4 cm for a smooth bottom and ii * --; lG_lj bottom. To estimate the condition when the lifting force becomes comparable with the reduced gravity force, Eqs. (16) and (17) can be used. This gives another critical value for the water depth: He*
=
2h,COS a(h*
p* - 1) l+K,
c,a:
1-p i
h*p*
Thus, for H,, 2 H * * (= 6-8 negligible effect.
5. Comparison
(34)
*’ +K,
I
h,), the bottom
with the results of laboratory
friction
can be considered
to have a
measurements
The solution (Eq. (28)) gives the cobble displacement x, for the case when E = I = 1 (h, < h, and the lift force remains less than the reduced gravity force on the cobble). In general, I, in accordance with Eqs. (16) and (17), depends on A = U - U, and may change its value during the cobble motion. In this case, standard numerical procedures can be used to solve Eq. (18) and to calculate the cobble displacements. Such -..1_..1_,:.%-,. ._.^__ -,.A- ..,:..lz,.” w,, IL\ \I,, 171 \I”,, /lr\\ \IL,, /i ?\ /I<\ (171 fin\ LUIU ,-,-I \A”,, 1-W\ all&l -..#l .Ltialculauu~ls WGIC:IIISLUG usmg Uqs. \IJ/) \r r,, 117, Ulci results were compared with the measured values of x, for all experiments. Some results of the calculations are shown in Figs. 6-8 (dashed lines). The variable external parameters were H,,, p,, D, h,, K and cr. Other physical parameters, K,, Cd, CL, g, p and u are constants as defined earlier. Taking into account typical errors of measurements, the agreement is satisfactory.
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60 60
I
I
I
/
I
I
I /
70 .,
60 ‘-
0 00
57
/
,
/
/
,
,d
/a
50 .-
Fig. 10. Calculated (x,) and measured (x,,,) cobble displacements (symbols) for 16 experiments conducted at h, > h, (cobbles were only partly covered by water). Smooth horizontal bottom, K = 0.1. Range of parameters: H,, = 14-23 cm, D = 10 cm, h, = 4, 5 and 6 cm, pC = 1.5, 1.7, 2.0 and 2.4 g cme3.
Similar comparisons were made for all experiments conducted at different values of the external parameters, and the results are shown in Figs. 10-12. In these graphs, the measured values of x,( x,,r) are plotted against the calculated values ( x,,,). For easier comparison, the results are presented in three different graphs. The data obtained for the case when h, > h, (cobble is not completely covered by water and the process is very sensitive to the water depth h, near the cobble) are shown in Fig. 10. These experiments were conducted using a smooth horizontal bottom and relatively small values of Ha. As
50 -F =
40-
f 30 --
30
40
50
60
I 70
80
xexp (cm)
Fig. 11. Calculated (I,,,) and measured (x,,,) cobble displacements (symbols) for 18 experiments conducted at h, < ho (cobbles were completely covered by water). Smooth horizontal bottom, K = 0.1. Range of parameters: Ho = 20-29 cm, D = 10 cm, h, = 4 cm, p, = 1.5 g cmm3.
58
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
Fig. 12. Calculated (x,) and measured (x,,,) cobble displacements h, < h, (cobbles were completely covered by water). Sandy sloping parameters: H,, = 20-26 cm, D = 10 cm, h, = 4 cm, pc = 1.5 g cme3.
for seven experiments conducted bottom, a = 2”. K = 0.4. Range
at of
can be seen in this graph, although the general agreement between the calculated xth and measured x,,,, values is satisfactory, the experimental points did not collapse along the 45” line and there is some scatter. This scatter likely arises from two sources. First, there is some scatter in the experimental data. For two runs conducted under presumably the same conditions, for example, variations in the measured results were often as great as the scatter exhibited in Fig. 10. Second, variations arise because our simplified model does not take into account all the important details of the cobble motion when h, > ha (e.g., splash of water, when the water front impinges on the cobble and subsequent changes of the water depth h, near the cobble). The data obtained for the case when h, < h, (cobble is completely covered by water and the process is less sensitive to the water depth h, near the cobble) are shown in Fig. 11 (smooth bottom) and in Fig. 12 (sandy bottom). Note that the results of the calculations are not very sensitive to the exact values of the drag and lift coefficients; e.g., when C,, or C, are changed by lo%, X, changes less than on 3%, as can be estimated from Eq. (28) where C, and C, are incorporated into B in accordance with Eq. (22).
6. Conclusions 72, 1116
-,A,... lll”Ll”ll
,.c
“I
I,...__ Icup
L,db.““l.L”lll
,....+:,.I,” paluuE;s
/_,.l.Ll,.“\ \L”““IG~,
:..
_
111 a
^._._ “I. SwaMI
“^_^
^_
^_
.!J”IK
“II
al,
:-.^^-~,.1-1~ IIqJ~~IIIc;d”IE;
sloping beach was modeled experimentally and analyzed theoretically. The purpose of the research was to better understand the fundamental dynamics of the interaction of the swash with cobbles initially resting on a sloping beach. In oceanic situations, the beach is, of course, permeable. Owing to the fact, however, that very little research has been done on this problem, it was decided to first investigate the more simplified case of an
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
59
impermeable bottom. This allowed one to consider a physical system with a reduced number of external parameters, but yet one which captured some of the realism of the oceanic situation. Even with this simplified problem, at least nine external parameters (H,, D, h,, pc, p, u, g, a and K) were shown to be important. In tbe present study, the impulsive action on a cobble of a unidirectional bore was considered as a basic element of the more complicated periodic swash characteristic of oceanic situations. In the experiments, a dam-break flow was used to simulate such an impulsive swash. It is shown that when the threshold for the initiation of motion is overcome, the cobble starts to move with relatively large velocity in the on-shore direction. The cobble displacements x, were measured as a function of time t and other external parameters. Based on the results of these experiments, a theoretical model was proposed and cobble displacements were calculated and compared with the measured values. Satisfactory agreement between the laboratory observations and the model predictions was obtained. In some sense, the model developed is rather general and permits incorporation of more complicated background conditions. Examples include the motion of cobbles in a periodic swash or in a wave-induced oscillatory flow as is characteristic of the shoaling region of the surf zone (Voropayev et al., 1998). On the other hand, the model proposed is rather simplified, and hence, does not take into account many important details which occur in the oceanic environment. For example, the complex motion field that is generated as the swash impinges on the cobble is not considered. Furthermore, the model assumes that the flow is spatially homogeneous in the fluid sheet, which is not strictly true in either the laboratory or the ocean. The model could also be improved by incorporating more realistic velocity distributions in spao? and time characteristic of oceanic swash zones. Improved parameterizations of the drag and lift on the cobble as well as the bottom friction are also desirable for improving the model. Finally, the extension of the model to include permeable beaches whose bedforms are free to change under the actions of currents and waves is also IWPA& _..I&& Cnrh tgrtr 21.p llUL nnt triwial need tn arlrlr~rwrl ;n nrrl‘w rl~xr,alnn hpttpr .lVVU”Y, LL&ULL” .&LV UI.IUI lwt “UL llVV.. L” h,a “V U..Ul”Y”I.., 111 “LUVl tn C” ““““F ““&Cl1 predictive
capabilities
for the motion of cobbles in oceanic swash zone environments.
Acknowledgements This research was supported by the Office of Naval Research, Grant N00014-1-9543. Useful discussions with Dr. T. Kinder (ONR), Lt. Commander C. Church (NRL) and Dr. T. Holland (NRL) are gratefully acknowledged.
References Arcilla, AS., Stive, M.J.F., Kraus, N.C. (Eds.), 1994. Coastal dynamics ‘94. Am. Sot. Civil Eng., NY. Batchelor, G.K., 1967. An Introduction to Fluid Dynamics. Cambridge Univ. Press, Cambridge. Battjes, l’.A., 1988. Surf-zone dynamics. Ann. Rev. Fluid Mech. 20, 257-293. Belorgey, M., Rajaana, R.D., Sleath, J.F.A. (Eds.), 1993. Sediment transport mechanisms in coastal environment:; and rivers. World Scientific, Singapore.
60
P.A. Luccio et al./ Coastal Engineering 33 (1998) 41-60
Cm&, A.D.D., 1985. Waves Interactions and Fluid Flow. Cambridge Univ. Press, Cambridge. Dalziel, S., 1993. Rayleigh-Taylor instability: experiments with image analysis. Dyn. Atmos. Oceans 20, 127-153. Eagleson, P.S., Dean, R.G., 1959. Wave-induced motion of bottom sediment particles. In: Fisher, J.S., Dolan, R. (Eds.), Beach Processes and Coastal Hydrodynamics. Dowden, Hutchinson and Ross, pp. 162-188. Engelund, F., Fred&, J., 1982. Sediment ripples and dunes. Ann. Rev. Fluid Mech. 14, 13-37. Fredsoe, J., Deigaard, R., 1992. Mechanics of coastal sediment transport. Advanced Series on Ocean Engineering, Vol. 3. World Scientific, Singapore. Klemn R Rntunnn ~kamnmck_._, WC.-., 1997. Gn me r__r-D-‘_-_. nrnnnpatinn of _.., i____..-_ -_ internal bores. J. F!ujd M_ech. 331, __.__..r, ”1.-., __“__...._, R 81-108. Longuet-Higgins, M.S., Stewart, R.W., 1964. Radiation stress in water waves: a physical discussion with applications. Deep-Sea Res. 11, 529-562. Lott, D.F., Poeckert, R.M., 1996. Extending cooperative research. Canada, New Zealand, United States in joint effort in British Columbia to evaluate penetrometers for ground-truthing acoustic classifiers for mine countermeasures. Sea Technol. 9, 56-61. Mei, CC., 1982. The Applied Dynamics of Ocean Surface Waves. Wiley, New York. Mei, C.C., Liu, P.L., 1993. Surface waves and coastal dynamics. Ann. Rev. Fluid Mech. 25, 2-240. Neilsen, P., 1992. Coastal bottom boundary layer and sediment transport. World Scientific, London. Peregrina, D.H., 1983. Breaking waves on beaches. Ann. Rev. Fluid Mech. 15, 149-173. Sleath, J.F.A., 1984. Sea Bed Mechanics. Wiley-Interscience, New York. Voropayev, S.I., Roney, J., Fernando, H.J.S., Boyer, D.L., Houston, W.N., 1998. On the motion of large l.~+t~m Iro..+;r*,o ;nrl.,,..%i nrr;,,.,tCx,.., r ./vuaUU mort01 Pnn rP.,iPI., U”LL”LLl ~~nrlr0 :,I, .TlllliP **LL”ti-IIIU”&tiU “&.ULU’“‘J on.., IgUn. a. YL’&, .,nAnr ULIYQIL1”lln. White, F.M., 1991. Viscous Fluid Flow. McGraw-Hill, New York. Whitham, G.B., 1955. The effects of hydraulic resistance in the dam-break 399-407. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York. Yalin, M.S., 1977. Mechanics of Sea Bed Transport. Pergamon, NY.
problem. Proc. R. Sot., Ser. A 227,
33 (1998) 41-60
The motion of cobbles in the swash zone on an impermeable slope P.A. Luccio a, S .I. Voropayev a,1,H.J.S. Fernando a, D.L. Boyer a,*, W.N. Houston b aEnvironmental Fluid Dynamics Program, Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 852874106, USA b Environmental Fluid Dynamics Program, Department of Ciuil and Environmental Engineering, Arizona State lJuiuersity; Tempe, AZ K5287-6106F USA Received 27 May 1997; revised 20 November
1997; accepted 26 December
1997
Abstract The purpose of this communication is to present the results of a series of laboratory experiments aimed at better understanding the dynamics of the motion of large bottom particles (cobbles) in a swash zone. In this region, a thin sheet of water that results from the collapse of a turbulent bore, runs up the beach and can induce the transport of relatively large solid objects in the on-shore direction. The aims of the study were to: (i) mimic this process in laboratory experiments and identify the associated physical processes invoived; and (ii) to deveiop a suitabie theoretical model to describe the motion of cobbles. The experiments employed a solid impermeable bottom and were conducted in a long tank of rectangular cross-section. An impulsive hydraulic bore, produced by a dam-break mechanism at one end of the tank, was used to simulate the water motion in the swash zone. Solid objects of simple discoid shape were used to model the cobbles. The results of the laboratory observations were compared with model predictions. In the range of external parameters used for the experiments (size and density of cobbles, propagation velocity and height of the water front, slope and friction at the bottom), a reasonable agreement between the measured and calculated values of the cobble displacement as a function of time was obtained. 0 1998 Elsevier Science B.V. Keyw0rd.r: Sediment transport;
Cobble dynamics
* Corresponding author. Tel.: + l-602-965-1382; fax: + I-602-965-1384; e-mail: [email protected]. ’ Also at the Institute of Oceanology, Russian Academy of Sciences, Moscow, 117851, Russian Federation. 0378-3839/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PIZ SO378-3839(98)00003-9
42
P.A. Luccio et al./ Coastal Engineering 33 (1998) 41-60
1. Introduction Sediment transport in the near-shore zone is of great importance for many engineering and geophysical applications. During the past several decades, extensive research has been conducted on the study of the nature and dynamics of surface waves and their interaction with the ocean floor (e.g., see Longuet-Higgins and Stewart, 1964; Engelund and Fredsoe, 1982; Peregrina, 1983; Craik, 1985; Battjes, 1988). The resulting knowledge has led to the development of sophisticated models of the wave-induced motion fields and accompanying transport of fine sediments under wave action (e.g., see Yalin, 1977; Mei, 1982; Mei and Liu, 1993; Sleath, 1984; Fredsije and Deigaard, 1992; Neilsen, 1992; Belorgey et al., 1993; Arcilla et al., 1994). Nonetheless, significantly less attention has been paid to the study of the motion of much larger objects, such as cobbles. The American Geophysical Union (e.g., Sleath, 1984) has classified cobbles as sediment particles with typical diameter D = 6.4-25.6 cm. Apart from obvious engineering and geophysical applications, the transport of such large particles as cobbles is a for anti-tank mines in the topic of current naval interest; i.e., devising countermeasures near-shore zone (e.g., see Lott and Poeckert, 1996). The near-shore zone may be divided into three subzones, wherein different mechanisms dominate the dynamics of the transport of bottom particles (e.g., see Battjes, 1988). As waves approach the beach parallel to the shore line, they induce oscillatory motions in the water column, thus producing unsteady forcing on particles resting on the bottom. These waves then either break or do not break. In the former case, the breaking waves generate a highly turbulent surf zone characterized by strong vortices and substantial vertical motion. The vortices can be sufficiently strong to detach rather large cobbles from the bottom. In the latter case, they collapse at the shore edge, without forming a surf zone. The breaking waves are characterized as being either plunging or spilling waves; further in shore, both types of waves eventually develop into a turbulent bore. As these hnrt=c nnnrnnrh the v”Lvu ..yy”-v-A .. . . wnt~rlim= ..I-v-l...l) thev “--, mllnnw v-*-“ry-
and ______ form IA I.. cwnch mn~ _-__ _Y__-----,
in yhi& __.
a &in s&et
of water resulting from the collapse runs up the sloping beach. In the swash zone, considered part of the surf zone, the beach is periodically covered by a relatively thin layer of water. The fluid motion here is strongly asymmetric: during the on-shore motion phase, the energy concentrates into a thin water sheet with a relatively sharp front which, when impinging on a cobble, may transmit significant amounts of momentum to the cobble, thus initiating its motion. The return phase, which occurs after the maximum wave run up, is controlled mainly by gravity and appears to be somewhat less energetic. Recognizing that in different regions of the near-shore zone, fundamentally different background motion fields determine the dynamics of the transport of large bottom particles, three separate experimental installations were constructed at Arizona State TT-:..___:r.. _&..,a..*L.. __1^_.^_& pal&K --~:,.1- &_ __^_^ _.&Lllt;LllillllSlllb. -,.,.L..-:“-‘p-,.,J-, “s~llla~“ly ,.“,.:ll..&--. “‘,,“t;lblLy 6,. L” bllllly LIK IeleVallL LlillqJ”IL I” III”Ur;l flow, a standing wave tank was constructed; to simulate the direct effect of breaking waves, a 33.5-m long tank in which progressive waves impinge on a sloping beach was developed; and to model the swash zone, as discussed herein, a dam-break tank was used. In the present paper, we report only on laboratory experiments obtained from the dam-break facility, which are akin to studies on cobble dynamics in swash zones.
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
43
The present work has been motivated by the desire to predict the motion of isolated large bottom particles (e.g., anti-tank mines) on smooth sloping beaches. It is noted that most natural cobbles in the oceanic environment are located in regions for which the overlying cobbles, which may be free to move, are resting on cobble beds. The motion of such (cobbles is a problem of interest but not the one to be addressed in the present communication. Furthermore, natural beaches are permeable with bedforms that change shape in the presence of currents and wave action. Nevertheless, owing to the relatively unexplored nature of research on the motion of large bottom particles in the oceanic environnnent, a reasonable starting point for research in this area is to consider an impermeable beach. Restricting to an impermeable bottom both simplifies the problem at hand imd, additionally, in due course, will aid in better understanding the dynamics of cobbles resting on a permeable sandy beach. In the present study, we restrict to cobble motions in the swash zone. Results of experiments with oscillatory flows akin to those conditions in the shoaling region (Voropayev et al., 19981, as well as for breaking waves, again with impermeable beaches, are to be reported separately. The remainder of the paper is organized as follows. The experimental installation and methods of study are described in Section 2. The results of the experimental observations are given in Section 3. A model is derived in Section 4 and comparisons with the experimental data are discussed in Section 5. Finally, the main conclusions are formulated in Section 6. 2. Experimental
installation and methods of study
As mentioned, a dam-break zone. In this approach, a dam of water, is instantaneously well-defined and reproducible rectangular cross-section. By h&&t onrl nrtxnom&nn cna=rl “U’~“C ULlLb yLvyY~uu”” 0yuAa
method was used to simulate the water motion in a swash or gate, which, prior to the experiment, blocks a reservoir or nearly instantaneously removed. This results in a bore which advects through the uniform channel of varying the height HO of the water behind the dam, the nf “I
the LllU hnre ““LU
r-an -cut
ho vu
~vGPA flZ*‘6. .c&LIUU \r
11 1,.
Two similarities can be noted between a breaking wave impinging on a beach and a turbulent bore; i.e., shape and speed. Apart from these large-scale visual similarities, there is also an internal dynamical similarity; namely, the energy lost through the dissipation in a bore gives realistic estimates of the energy lost in the surf zone (e.g., see Fredsije and Deigaard, 1992). These similarities are the principal reasons the dam-break method was chosen to model the water motion in a swash zone.
6)
Fig. 1. Schematic diagram of (a) the experimental -cobble, 4-gate, 5-water.
@I set-up and (b) model cobble; l-tank,
2-sloping
bottom, 3
44
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
The experiments were conducted in a rectangular Plexiglas tank having a length of 250 cm, a height 60 cm and a width 30 cm (Fig. 1). A vertical removable Plexiglas gate was installed at a distance of 60 cm from the end of the tank and this smaller section was tilled to the desired level (H,, = 5-30 cm) with slightly-dyed tap water of room temperature (density p = 1 g cme3, kinematic viscosity v= lo-’ cm* s-l>. Two separate tank floors were used in the experiments; i.e., smooth (Plexiglas) and rough, for which a thin layer of sand was glued to the Plexiglas. This permitted a variation of the kinetic friction coefficient K between the bottom and cobbie by a factor of four. The values of K (0.1 f 0.01 and 0.4 + 0.04) were determined directly by slowly dragging a model cobble in still water and using standard dynamometric methods to measure the resulting friction force. By tilting the bottom at a small angle cx to the horizontal, the effect of a sloping bottom was produced in the experiments. Model cobbles can be characterized by a diameter D, a mean height h, and density p,. The model cobbles of discoidal shape, with rounded comers of radii = 0.5 cm, D = 10 cm and h, = 4-6 cm, were hollow and were made using wooden forms; the materials used for the outer shell were a polyester resin and fiberglass. By filling the model cobbles with different materials (sand or polystyrene or glass spheres), it was possible to change the mean cobble density p, from 1.5-2.4 g cmp3. In the experiments, one of the cobbles was placed at a fixed distance (50 cm) from the gate in the empty part of the tank. To initiate an experiment, the gate was then rapidly removed vertically. The side view of the cobble motion was recorded using a super VHS video camera located at a large distance (4-5 m) from the tank; this assured that parallax errors would be minimized. In some cases, a standard 35-mm camera was also used. The video tape was then digitized frame by frame using commercially available digital processing software (DigImage: Dalziel, 1993) and all experimental data (positions of the cobble and height and positions of the water front) were obtained from these digitized images using standard data processing techniques. Each experiment lasted only l-2 s and, without such rapid digitizing procedures, the errors in measurements would Inroe rhp tvniml in -__ rnhhk __--__ lnrdnn ___- ________ ,._-__ have __I._ hem v ____ tnn ___ _-p” ‘, r __ I_ errnr - -- _ - in ___ &em_inip_v e the time and space from the measured values of the instantaneous cobble displacement x,(t) using DigImage was approximately 2-3%. A total of 41 experiments were conducted; the results of these experiments are discussed in Section 3.
3. Results of observations This section will be used to provide a discussion on the background present qualitative observations of bore/cobble interactions.
flow and to
3. I. Background flow Consider the background flow produced by the dam-break method. It is useful to note that the classical solution of this problem, derived by Bitter in 1892 (e.g., see Whitham,
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
b
0
Fig. 2. Slcetch of the dam-break flow. Solid line-water bottom friction at the tip of the front.
52
4.5
x+
surface without friction, dashed line-correction
for
19741, has no bore (Fig. 2). After the gate, which is at x = 0, is opened at t = 0, the water surface takes a parabolic shape with the height h (x, t) given by:
h=-f- 2( gH,)1’2 9g [
and the fluid particle velocity u=3
2r
(gH0)1/2+fl, L
1
x 2
- ;
(1)
u along the x-axis given by:
(2)
I
where g is the gravitational acceleration, H,, is the initial water height behind the gate I’* _< w/w < 2(gH,) ‘I2 . The front of the flow, at x = x+, has zero and -(g&J height (h, = 0) and propagates in the positive direction along the x-axis with the velocity u + equal to the fluid particle velocity u at this point:
dx+
u+= dt
(3)
= 2( gH,)“*.
The point where h = H, propagates
with the velocity:
. ._
dx_
‘-= - dt = -(
in the opposite direction
gffoy,
(4)
while: k!=%H,andu=c(gH,)l,2atx=O.
(5)
In a real flow (e.g., which will be depicted in Fig. 5a below), the height h, of the tip of the front is finite (Fig. 2) and propagates with the velocity u0 = dx,/dt, which is less than u+. These differences were explained by Whitham (1955), who considered frictional resistance near the tip of the front, resulting from turbulence, which dominates the flow dynamics in this region. His modified boundary layer solution shows that h,, as well as uO, remains approximately constant through the tip of the front and: /I, = a,H,,
(6)
where a2 is a slowly increasing function of time, with a2 = l/4 at small times. This prediction is in very good agreement with our measurements shown in Fig. 3. The values of h, in all experiments conducted at different values of H,, were taken at the
P.A. Luccio et al. /Coastal
46
Engineering 33 (1998) 41-60
Fig. 3. Height, ho, of the water front as a function of water height Ho all experiments were taken at a distance 20 cm behind the front, when from the gate. Symbols-experimental data: (0) smooth horizontal dashed lines-approximation (Eq. (6)) with a2 = 0.2 (smooth bottom) measurement errors are given by the vertical bars.
behind the gate. The the front was at the bottom, (0) sandy and a2 = 0.3 (sandy
values of ho for point x = 50 cm sloping bottom; bottom). Typical
moment when the front reached the point x = 50 cm and the height h, of the front was measured at the same distance (20 cm) behind the leading edge of the front. Although the coefficient a2 increases when the bottom friction increases, for the range of H, used in the experiments the results of measurements can be approximated by the simple dependence (Eq. (6)) with: a2
=
constant = i
0.2, smooth bottom, 0.3, sandy bottom,
and this parameterization is shown by the dashed lines in Fig. 3. Whitham (1955) also predicted that at small times, the frontal rapidly from u + = 2( gH, )I/* to the asymptotic value:
(7)
velocity
ua drops
where a, = 2/3 at intermediate times. This is also in agreement with the results of our measurements shown in Fig. 4. Using the data on the position of the front x,, as a function of time t, mean values of the frontal velocity IQ, = dx,/dt, were calculated for small times (at x 5: 50 cm) and are shown in normalized form in Fig. 4 for experiments __-XL._r_J a~ ..&UIIIE;ICIIL XCc^-n-c _._I ..^^ VI -C no. u -_ .I&_UI ^CIL__.. ^^^.__^-^-._ give -:..- L?,r 1.4 fCK CVII~~CL~U villues rut; _^^. IGSUILS ulest: -~~~t;asuro~ueu~s both smooth and sandy bottoms (see Fig. 4). From the general theory of internal bores arising in two-layer fluids, it follows that in the limiting case of an external bore (one fluid layer), a, and a2 are related by a: = (1 + a,)/2a, (see relation (4) in Klemp et al., 1997). This gives the estimates: a, = 1.7 for a2 = 0.2 and a, = 1.5 for a2 = 0.3, which are in reasonable.agreement with the measured values.
P.A. Luccio et al./ Coastal Engineering 33 (1998) 41-60
n *I 0
1.5 -
e
00
-c--_-------o~-o0
0.0
a,=1.4 o---,,
n
n
a I
d
l-
$
0.5 -
01
0
I
I
I
I
I
I
5
10
15
20
25
30
Ho(cm) Fig. 4. Non-dimensional front velocity, u0 /(gH,) ‘I2 at a small distance (X = 50 cm) from the gate for different values of If,,. Symbols-experimental data: (0) smooth bottom, (0) sandy bottom. Dashed line-approximation (Eq. (8)) with a, = 1.4. The values of u. = dxO /d t werecalculated using the experimental data on the front position x0 as a function of time; a typical error bar is shown.
The fluid particle velocity u behind the front is less than the frontal velocity larger than the fluid velocity at x = 0, thus:
u0 but
(9) Recognizing that a, drops rapidly to the asymptotic theoretical the following, the fluid velocity u is approximated by: 24 =
Ul( gfl,y,
value with a, = 2/3,
in
(10)
where ‘zl = constant = 2/3. 3.2. Motion of the cobbles In the experiments, the cobbles were placed at a fixed distance (n = 50 cm) from the gate along the tank axis (Fig. 5a). The gate was opened and the water front was created. The distance from the gate to the initial position of the cobble was the same (50 cm) in all _-___. mns. -_
Thin ___-I dictnnre - _-_____
wa< ..-- chnnen ___- -___ emnirirnllv ____r____-__,
in ~r&r
TV nmvide r--.---
R tin of qprgxi~_gte!y - -*r
constant height as the water front reached the cobble. Upon impingement of the water front (Fig. 5b), the cobble started to move (Fig. 5c) with a velocity U = dx,/dt, where X, is the cobble displacement from the initial position, x, = 0 (x = 50 cm) at t = 0. Typic&/ plots of the cobble displacement x, as a function of time t for runs with different initial conditions are shown for a smooth horizontal and a sandy sloping bottom in Figs. 6 and 7, respectively. In these data, the time origin t = 0 was chosen at the moment when the cobble started to move; the cobble displacement was measured from the initial position of the cobble x, = 0 at t I 0.
48
P.A. Luccio et al. / Coastal Engineering 33 (I 998) 41-60
Fig. 5. Sequence of photographs showing the motion of a cobble in a dam-break flow: (a) top view, (b-c) side view; (a) t < 0, (b) t = 0, (c) t = 0.04 s, (d) t = 0.67 s. The flow moves from right to left. The time origin, t = 0, is chosen at the moment when the cobble, which was initially at rest at x = 50 cm, starts to move. Experimental conditions: H, = 20 cm, D = 10 cm, h, = 4 cm, p, = 1.5 g cm13; smooth horizontal bottom.
As can be seen in these figures, the cobble starts to move with some finite initial velocity, U, # 0 at t = 0; i.e., the cobble motion is an impulsive one. This initial velocity is relatively large and depends on the initial conditions of the experiment, but remains significantly less than the mean velocity U of the water sheet. As a result, the
49
P.A. Luccio et al. /Coastal Engineering 33 (1998) 41-60
Fig. 6. Typical experimental data (symbols) showing the cobble displacement xc as a function of time t for a smooth he3rizontal bottom. The time origin t = 0 is chosen at the moment when the cobble, which was initially at X, = 0 (X = 50 cm), starts to move. Dashed lines-the model estimates (see below). Experimental conditions:(+)H~=18cm,D=10cm,h,=6cm,p,=2.0gcm~3;(~)H~=20cm,D=10cm,h,=5 cm, pC = 2.4 g cm ~3;(~)H~=20cm,D=10cm,h,=4cm,p,=1.7gcm~3;(O)H~=29cm,D=10 cm, h,=6cm, pC=1.7gcm-3.
water front passes the moving cobble and continues to move with approximately constant velocity ahead of the cobble (Fig. 5~). For the case of the smooth horizontal bottom, the cobble velocity U gradually increases and reaches an approximately steady
1.6
Fig. 7. Typical experimental data (symbols) showing the cobble displacement X, as a function of time t for sandy sloping bottom. Dashed lines-the model estimates (see below). Experimental conditions: (0) Ho = 20 cm, D = 10 cm, h, = 4 cm, pC = 1.5 g cmu3; (*)~,,=23cm,D=10cm,h,=4cm,pC=1.5gcm-3;(0) H,=29cm, D=lOcm, h,=4cm, p,=1.5gC11-~.
50
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
value at t = l-l 3 s (Fig. 6). For the sandy sloping bottom, U remains constant or decreases slightly with time (Fig. 7). At larger times (not shown in Figs. 6 and 71, because of the finite length of the tank, the water front reaches the end wall of the tank, reflects and starts to move in the opposite direction. This leads to a rapid decrease of the cobble velocity, and in some runs conducted at relatively large values of the water height H, behind the gate, a reverse motion of the cobble was observed. To eliminate the influence of the reflected motion, only the data when the front moved from the gate to the end wall were used and compared with the estimates predicted by the model. As can be seen in Figs. 6 and 7, the cobble displacement for the same time, say _x: at t = 1 s, depends strongly on the initial conditions and changes by a factor of four from _x: = 20 cm to .x: = 80 cm. The values of xi, as might be expected, increase for increasing H, and decreasing pc. One of the interesting observations is that the bottom frictional coefficient K does not influence significantly the initial cobble motion. This can be seen clearly in Fig. 8, where cobble displacements x, are shown for two cases for which all experimental conditions except for bottom friction were the same; i.e., K = 0.1 (smooth bottom) and K = 0.4 (sandy bottom). In spite of this relatively large change in the value of K for the two experiments, the resulting change in the cobble displacement with time is less than 20%; see, for example, t = 1 s, showing that bottom friction does not influence significantly the dynamics of the cobble. Detailed visual observations show that at relatively high propagation velocities, the cobbles may be detached, at least partly, from the bottom and this effect may significantly reduce the influence of bottom friction. This is not to say that the value of K does not influence significantly the initiation of cobble movement. Recall that both n, and t are zero until the cobble movement is initiated.
i.:
Fig. 8. Cobble displacement xc as a function of time t for experiments for which all parameters were the same, except the bottom friction coefftcient, K: (0) smooth bottom (K = 0. l), ( + ) sandy bottom (K = 0.4). Other parameters: Ho = 23 cm, D = 10 cm, h, = 4 cm, pc = 1.5 g cmm3, (Y= 0”. Dashed lines-model estimates: upper-for smooth bottom, lower-for sandy bottom.
P.A. Luccio et al,/ Coastal Engineering 33 (1998) 41-60
51
A total of 41 experiments were conducted with cobbles using different values of the external parameters Ha, p,, h,, D, a and K, where Ha is the water depth behind the gate, p,,, h, and D are the cobble density, height and diameter, (Y is the bottom slope and K is the kinetic bottom friction coefficient. In addition, 16 experiments were conducted without cobbles to obtain the data on the background flow characteristics (Figs. 3 and 4). In the experiments with cobbles, some threshold value of Ho must be exceeded to initiate the cobble motion (see the estimates below), and in these runs, H, was varied in the range H, = 9-29 cm. The values of the other parameters were set as follows: p, = 1.5, 1.7, 2.0 and 2.4 g cmm3, h, = 4, 5 and 6 cm, D = 10 cm, cx= 0 and 2” and K = 0.1 (smooth bottom) and 0.4 (sandy bottom). A model for cobble motions was developed concurrently with the experimental studies and is presented in Section 4. The proposed model can be used to calculate the cobble displacement X, as a function of time t and other external parameters. The results of these calculations are compared with the experimental data.
4. A model and some estimates The model to be described here was developed essentially from basic principals while incorporating certain parameterizations based on present and past empirical results. The system under consideration is shown schematically in Fig. 9. The cobble is initially (t I 0) at rest, with its center at X, = 0. A water sheet of depth h and density p propagates with velocity U along the slope and at the moment t = 0 impinges on the cobble. During the impact, the water transmits some momentum to the cobble and, if
Fig. 9. A schematic of the model flow configuration: (a) side view, (b) top view. Incoming water front propagates from left to right with velocity E(t) along the sloping bottom and at t = 0 impacts initially still cobble, which is at x, = 0. During the impact, the water transmits some momentum to the cobble which is initially at rest and at t > 0, the cobble moves with the velocity U(f) = dx, /dt, which changes with time, because of the bottom friction and positive (U < 7i) drag force.
52
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
this momentum exceeds a threshold value, the cobble starts to move with some velocity U,, in the positive direction of the x-axis, which is parallel to the bottom. Owing to bottom friction and the drag force, the propagation velocity of the cobble U = dx,/dt changes with time t. We wish to determine the dependence of the cobble displacement X, from its initial position on the time elapsed and other external parameters. To derive the governing equation, assume that the fluid everywhere in the neighborhood of the cobble would have had the velocity U in the absence of the cobble. This is the standard assumption used to describe
(11)
In deriving this expression, it was assumed that the acceleration reaction (Batchelor, 1967) on the cobble is equivalent to MK, d/dt(U - U), where M is the mass of water displaced by the cobble and K,( = 1) is the virtual mass coefficient. The total inertia M,(dU)/(d t) + MK,d/dt(lJ - U) is thus assumed to be balanced by the buoyancy contribution G sin (Y, where G = (M, - M)g, the pressure gradient contribution M(dE)/(dt), the total drag force on the cobble Fd and the bottom friction force Ff. It is expected that the pressure gradient force on the cobble is the same as that on an equivalent fluid mass, which, for the inviscid limit, is the same as M(dZ)/(dt). Denote p * = p,/p, A = ii - U,
(12) and use M = (rr)/(4h tions: 1 r”d = zps,(ii Ff=
* )ph, D2, M, = (~)/(4)p~
h, 0’
and the standard parameteriza-
- Uj2C&
I.“\ (‘3)
a-
(14)
-lK(Gcos
EF,);
here, S, = h, D/h * is the cobble cross-sectional area as it is immersed in the fluid, C, = C,(Re) is the drag coefficient, Re = (U - U)D/u is the instantaneous Reynolds number of the cobble, K is the kinetic friction coefficient at the bottom,
FL = ;pS,(ii
- U)‘C,
(16)
is the lifting force arising when h, < h,, h * = 1, C, = C,(Re) is the lifting coefficient, S, = TD~/~ and: z=
EF,,
1,Gcos
a>
0,Gcos
CYIEF~.
(17)
53
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
In derivmg Eq. (15) it was assumed that when h, > h,, the principal contribution to the lift force is the drop of pressure along the top of the cobble where the water velocity relative to the cobble exceeds the water velocity away from the cobble. When h, < h,, it was assumed that there is no water above the cobble and, as a result, no drop of pressure. This leads to the parameterization (Eq. (15)). Relation (15) assumes that when h * > 1, h, > h,, there is no lifting force, and Eq. (17) shows that the bottom friction Ff become;3 negligible when the lifting force exceeds the buoyant weight of the cobble. Eq. (11) can now be written in the form:
(18) To solve Eq. (18) it is necessary to specify the functions C,(Re), and C,_(Re) and the water motion which can be characterized by the mean velocity U and depth h, of the water sheet. Let the water sheet propagate along the sloping bottom as a slab of thirknw:c h. = mnctant~ renrewntd hv. ...A~~y.~l,”“” vV”“.-.r) the= .&._ velnritv . v’.,.?“J ran w-1 he -1 ‘“y’w”“““.. v-7. ii = u - gt sin ff ,
(1%
where Eqs. (6) (7) and (10) can be used for h, and u. In deriving Eq. (19) the action of the gravity force on the water sheet on the sloping bottom was taken into account. In the absence of the bottom, the drag coefficient C, for a round cylinder moving in a large volume of water with velocity A = S - U, with good accuracy can be approximated as C, = 1.0 for lo* I Re I 2 X lo5 (see, e.g., White, 1991). In the present experiments, this corresponds to the range: 0.1 cm s-l I A s 200 cm SK’, where C, may be considered as a constant (D = 10 cm, u= lo-* cm* s-’ were used for this estimat’e). In the presence of the bottom, the drag coefficient may change. For example, *l__ ?I___ ___~A-:_:__rc__ _ __L___ :________ L_. ^ C--r_.. _l? _---__:--r-l_. 1 c -I l- *l__ UK urag C”~lllCltx,L 101 a spert: IIIl.xcils~Y uy a la~lul WI appruxmkmxy I .J--L III UK presence of the bottom. But it can be shown (see, e.g., Eagleson and Dean, 1959) that such an increase of C, for a sphere can be attributed at least partly to the additional friction because of the rotation of a sphere rolling along a sloping bottom. We acknowledge that in general, the coefficients C, and C, are complicated functions of geometry and other external parameters, with typical values C,, C, = 0.5-l (see, e.g., Yalin, 1977). For the geometry of the present application (solid bottom, finite depth of fluid), exact values of C, and C, are unknown and for simplicity these coefficients were approximated as follows:
(20)
C, = c, = constant = 1 .O, which were used to solve Eq. (18j and to derive quantitative motions. For U given by Eqs. (19) and (18) for t > 0, becomes: dA itdt + BA* = C,
estimates
of cobble
(21)
54
P.A. Luccio et al./ Coastal Engineering 33 (1998) 41-60
where for brevity: A=(p*h*
+K,),
(22) c = ZgK( p * h * are lused .
Tha lllcl
- 1)cos CY,
anlnt;nn LI”lULl”ll
nf “I
E’n Yy.
i31i \tiI,
.wh;,-h nlllrll
cca~ the rnn&t;nn ilcaCI0IIUD Cl&cl ~“IIUIII”II
A z Y -
A Y()
.zt b t=~ UC
n “+,
P’J” tic&II
be easily found and, for example, for E, Z = 1: I
_ Cl/2
B’/2A,
1-t
e(-2B’/2C”Z/A)r B’/2A
0
B1/2A
1+
+
Cl/2
_
Cl/2
(23) (-2B”2C”2/A)t
Bl/~~
+
Cl/2
\
To calculate the initial value of A,, = U - U,,, we must estimate the velocity U,, of the cobble just after (t = 0,) the impact of the water front, which has at the moment of impact (t = 0) a velocity U = u given by Eqs. (10) and (19). To estimate U,, one may ~~_~ IL- r_-rl- __. .wltn ..CL1.r,.. .___--_1 f--.-.-_r _l? _ _-If> L-J_. ._.L1_L -_- _.^^ ._.:*I_ Al__ _._I__:*_. use me dndqy me nunndl impdu 01 a xuiu oouy, wmm IIIUVCS WIUI LIIGve~ocuy U, on a free surface of liquid. Suppose, that at t < O_ the water sheet is at rest and its front (vertical free surface in our case) is approximately at the center of the cobble (see Fig, 9b). At the moment t = O_, the water starts to move and during the short time interval, O_ I t 4 O,, its velocity increases from zero to u. In the ideal case, this process can be described by a singularity similar to the delta-function. For this short time interval, the leading term on the right side of Eq. (11) is the first term. Integrating Eq. (11) over the short time interval, O_ I t I O,, and neglecting the other small terms on the right hand side, we obtain an estimate for the initial velocity of the cobble:
u =p 0
(1+4n) (p*h*
-H&J
u,
where the coefficient /3 < 1 is introduced to take into account that the cobble starts to move into free space, where there is no liquid at the moment of impact. It can be shown (e.g., see Batchelor, 1967) that: p = constant = l/2 for bodies of simple geometry. 2
(25) ’
In deriving Eq. (24), several effects were neglected: i.e., (i) the splashing of water when the front impinges on the cobble, (ii) the presence of the solid bottom on the lower side of the flow and the free surface above, and (iii) bottom friction. It is not at all clear how to include all of these very complicated factors into the physicai modei; aitemativeiy, it is preferred to use the simpie estimate (Eq. i24)) to derive realistic vaiues for Us. In fact, the coefficient p in Eq. (24) cannot be derived analytically for the present case; neither can it be considered as a universal constant. The average value p = 0.5 was chosen empirically by comparison of the measured (using the slope dx, /dt at small t) and the estimated values of Ua. Only by chance did this value of /3 coincide with Batchelor’s estimate (Eq. (25)). Some justification for this simplification is the fact that at least in the range of parameters used in the present experiments, only one value for p had to be used (see below).
55
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
Thus, taking into account that the cobble velocity is:
we have for the cobble displacement:
(27)
x,=jnfIidf-/nlAdr,
where the first integral gives the position of the water front and the second gives the distance from the water front to the center of the cobble, which is behind the front for t > 0. Csing Eq. (19) for U and Eq. (23) for A, the integrals in Eq. (27) can be calculated and the cobble displacement x, can be presented in closed form: I @B
1 x,:=ut--ggt*sina+ 2
1/2C”2 /A)t
C II2 A Vn i B i
_
(BYA,
-
(BW,
+ P)
1 _ (B1’2do(B%,
\
~7~‘~)
P)
’
’
+ Cl’*) (28)
where A, B and C are determined by (24). Before comparing model estimates consider certain limiting case. Assume the cobble are negligible (i.e., C,, K = limit B, C + 0 in Eq. (28), one obtains
Eq. (22), A, = U - U,, and U,, is given by Eq. with the experimental data, it is instructive to that the viscous drag and the bottom friction on 0) from which B, C = 0 in Eq. (22). Taking the for the cobble displacement:
1 lim x, = U,t-Tgt2sinff.
(29)
k3,C-t0 ~XTI__-
WIltm
..-1..
0111y
Al__
l__n_-
Lilt: UULLUIIl
c:_r:__ IIICLIUII
_Al__ 011 UK
__l_l_l_
CUIJDlC
:_
-__,:_:l_,_
IS Ilegllglulc,
---
UIlt:
c-_a_. 1IIIuS;
1 lim x, =ui---gr2sina-~ln(l+~Aor). C+O
For the case when both the drag and lifting forces are unimportant: 1 C lim x, = U,t-Sgt’sincY-xt’coscu. B+O
(31)
At small times, Eqs. (30) and (31) coincide with Eq. (29) and the cobble moves with the initial velocity, U,, in all three cases. At larger times, two effects begin to play a role: (i) acceleration (the last term in Eq. (30), which must be expressed to an accuracy of rr/.?\\ 1_ -P *l .. . _~. ufr-~~: Decause or me posiuve viscous drag (U < u), and (ii) deceieration (the iast term in Eq. 1(31)), because of bottom friction. For typical values of the parameters used in the experiments, these terms are comparable at intermediate times, and one may expect that there will be some balance between these terms. Thus, the drag, as well as bottom friction (and hence, the lifting force) are important and neither of these effects can be neglecied in the complete solution (Eq. (28)).
56
P.A. Luccio et al./Coastal
Engineering 33 (1998) 41-60
It is clear from the physics that to initiate the motion of the cobble, the value of H, must exceed some threshold value, say H * . This threshold may be estimated as follows. To calculate U,, Eq. (11) was integrated over a small time interval, and the last two terms in Eq. (11) were neglected. This approximation is correct for relatively large values of H,,, when the first term in Eq. (11) exceeds the friction term. By estimating the time of impact as r = D/u and integrating Eq. (11) over this time interval, it is possible to obtain an approximate criterion for the initiation of motion (dU/dt > 0): P(l+K,)Mu=(M,--M)gr(sin(~+K,coscu),
(32)
and this gives: (huh*-l)D H*
z
(sincr+K,coscu), (1+
(33)
K,)M
K is the static friction coefficient where Eq. (10) was used for u and K, = (1.52), (when the cobble is not sliding). In deriving Eq. (33), it was assumed that during the impact Fd and FL, which enters into Ff in accordance with Eq. (14), can be neglected. For the typical range of parameters used in the experiments, Eq. (33) gives the .. cm for a sandy estimates: ii * = 2-4 cm for a smooth bottom and ii * --; lG_lj bottom. To estimate the condition when the lifting force becomes comparable with the reduced gravity force, Eqs. (16) and (17) can be used. This gives another critical value for the water depth: He*
=
2h,COS a(h*
p* - 1) l+K,
c,a:
1-p i
h*p*
Thus, for H,, 2 H * * (= 6-8 negligible effect.
5. Comparison
(34)
*’ +K,
I
h,), the bottom
with the results of laboratory
friction
can be considered
to have a
measurements
The solution (Eq. (28)) gives the cobble displacement x, for the case when E = I = 1 (h, < h, and the lift force remains less than the reduced gravity force on the cobble). In general, I, in accordance with Eqs. (16) and (17), depends on A = U - U, and may change its value during the cobble motion. In this case, standard numerical procedures can be used to solve Eq. (18) and to calculate the cobble displacements. Such -..1_..1_,:.%-,. ._.^__ -,.A- ..,:..lz,.” w,, IL\ \I,, 171 \I”,, /lr\\ \IL,, /i ?\ /I<\ (171 fin\ LUIU ,-,-I \A”,, 1-W\ all&l -..#l .Ltialculauu~ls WGIC:IIISLUG usmg Uqs. \IJ/) \r r,, 117, Ulci results were compared with the measured values of x, for all experiments. Some results of the calculations are shown in Figs. 6-8 (dashed lines). The variable external parameters were H,,, p,, D, h,, K and cr. Other physical parameters, K,, Cd, CL, g, p and u are constants as defined earlier. Taking into account typical errors of measurements, the agreement is satisfactory.
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60 60
I
I
I
/
I
I
I /
70 .,
60 ‘-
0 00
57
/
,
/
/
,
,d
/a
50 .-
Fig. 10. Calculated (x,) and measured (x,,,) cobble displacements (symbols) for 16 experiments conducted at h, > h, (cobbles were only partly covered by water). Smooth horizontal bottom, K = 0.1. Range of parameters: H,, = 14-23 cm, D = 10 cm, h, = 4, 5 and 6 cm, pC = 1.5, 1.7, 2.0 and 2.4 g cme3.
Similar comparisons were made for all experiments conducted at different values of the external parameters, and the results are shown in Figs. 10-12. In these graphs, the measured values of x,( x,,r) are plotted against the calculated values ( x,,,). For easier comparison, the results are presented in three different graphs. The data obtained for the case when h, > h, (cobble is not completely covered by water and the process is very sensitive to the water depth h, near the cobble) are shown in Fig. 10. These experiments were conducted using a smooth horizontal bottom and relatively small values of Ha. As
50 -F =
40-
f 30 --
30
40
50
60
I 70
80
xexp (cm)
Fig. 11. Calculated (I,,,) and measured (x,,,) cobble displacements (symbols) for 18 experiments conducted at h, < ho (cobbles were completely covered by water). Smooth horizontal bottom, K = 0.1. Range of parameters: Ho = 20-29 cm, D = 10 cm, h, = 4 cm, p, = 1.5 g cmm3.
58
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
Fig. 12. Calculated (x,) and measured (x,,,) cobble displacements h, < h, (cobbles were completely covered by water). Sandy sloping parameters: H,, = 20-26 cm, D = 10 cm, h, = 4 cm, pc = 1.5 g cme3.
for seven experiments conducted bottom, a = 2”. K = 0.4. Range
at of
can be seen in this graph, although the general agreement between the calculated xth and measured x,,,, values is satisfactory, the experimental points did not collapse along the 45” line and there is some scatter. This scatter likely arises from two sources. First, there is some scatter in the experimental data. For two runs conducted under presumably the same conditions, for example, variations in the measured results were often as great as the scatter exhibited in Fig. 10. Second, variations arise because our simplified model does not take into account all the important details of the cobble motion when h, > ha (e.g., splash of water, when the water front impinges on the cobble and subsequent changes of the water depth h, near the cobble). The data obtained for the case when h, < h, (cobble is completely covered by water and the process is less sensitive to the water depth h, near the cobble) are shown in Fig. 11 (smooth bottom) and in Fig. 12 (sandy bottom). Note that the results of the calculations are not very sensitive to the exact values of the drag and lift coefficients; e.g., when C,, or C, are changed by lo%, X, changes less than on 3%, as can be estimated from Eq. (28) where C, and C, are incorporated into B in accordance with Eq. (22).
6. Conclusions 72, 1116
-,A,... lll”Ll”ll
,.c
“I
I,...__ Icup
L,db.““l.L”lll
,....+:,.I,” paluuE;s
/_,.l.Ll,.“\ \L”““IG~,
:..
_
111 a
^._._ “I. SwaMI
“^_^
^_
^_
.!J”IK
“II
al,
:-.^^-~,.1-1~ IIqJ~~IIIc;d”IE;
sloping beach was modeled experimentally and analyzed theoretically. The purpose of the research was to better understand the fundamental dynamics of the interaction of the swash with cobbles initially resting on a sloping beach. In oceanic situations, the beach is, of course, permeable. Owing to the fact, however, that very little research has been done on this problem, it was decided to first investigate the more simplified case of an
P.A. Luccio et al. / Coastal Engineering 33 (1998) 41-60
59
impermeable bottom. This allowed one to consider a physical system with a reduced number of external parameters, but yet one which captured some of the realism of the oceanic situation. Even with this simplified problem, at least nine external parameters (H,, D, h,, pc, p, u, g, a and K) were shown to be important. In tbe present study, the impulsive action on a cobble of a unidirectional bore was considered as a basic element of the more complicated periodic swash characteristic of oceanic situations. In the experiments, a dam-break flow was used to simulate such an impulsive swash. It is shown that when the threshold for the initiation of motion is overcome, the cobble starts to move with relatively large velocity in the on-shore direction. The cobble displacements x, were measured as a function of time t and other external parameters. Based on the results of these experiments, a theoretical model was proposed and cobble displacements were calculated and compared with the measured values. Satisfactory agreement between the laboratory observations and the model predictions was obtained. In some sense, the model developed is rather general and permits incorporation of more complicated background conditions. Examples include the motion of cobbles in a periodic swash or in a wave-induced oscillatory flow as is characteristic of the shoaling region of the surf zone (Voropayev et al., 1998). On the other hand, the model proposed is rather simplified, and hence, does not take into account many important details which occur in the oceanic environment. For example, the complex motion field that is generated as the swash impinges on the cobble is not considered. Furthermore, the model assumes that the flow is spatially homogeneous in the fluid sheet, which is not strictly true in either the laboratory or the ocean. The model could also be improved by incorporating more realistic velocity distributions in spao? and time characteristic of oceanic swash zones. Improved parameterizations of the drag and lift on the cobble as well as the bottom friction are also desirable for improving the model. Finally, the extension of the model to include permeable beaches whose bedforms are free to change under the actions of currents and waves is also IWPA& _..I&& Cnrh tgrtr 21.p llUL nnt triwial need tn arlrlr~rwrl ;n nrrl‘w rl~xr,alnn hpttpr .lVVU”Y, LL&ULL” .&LV UI.IUI lwt “UL llVV.. L” h,a “V U..Ul”Y”I.., 111 “LUVl tn C” ““““F ““&Cl1 predictive
capabilities
for the motion of cobbles in oceanic swash zone environments.
Acknowledgements This research was supported by the Office of Naval Research, Grant N00014-1-9543. Useful discussions with Dr. T. Kinder (ONR), Lt. Commander C. Church (NRL) and Dr. T. Holland (NRL) are gratefully acknowledged.
References Arcilla, AS., Stive, M.J.F., Kraus, N.C. (Eds.), 1994. Coastal dynamics ‘94. Am. Sot. Civil Eng., NY. Batchelor, G.K., 1967. An Introduction to Fluid Dynamics. Cambridge Univ. Press, Cambridge. Battjes, l’.A., 1988. Surf-zone dynamics. Ann. Rev. Fluid Mech. 20, 257-293. Belorgey, M., Rajaana, R.D., Sleath, J.F.A. (Eds.), 1993. Sediment transport mechanisms in coastal environment:; and rivers. World Scientific, Singapore.
60
P.A. Luccio et al./ Coastal Engineering 33 (1998) 41-60
Cm&, A.D.D., 1985. Waves Interactions and Fluid Flow. Cambridge Univ. Press, Cambridge. Dalziel, S., 1993. Rayleigh-Taylor instability: experiments with image analysis. Dyn. Atmos. Oceans 20, 127-153. Eagleson, P.S., Dean, R.G., 1959. Wave-induced motion of bottom sediment particles. In: Fisher, J.S., Dolan, R. (Eds.), Beach Processes and Coastal Hydrodynamics. Dowden, Hutchinson and Ross, pp. 162-188. Engelund, F., Fred&, J., 1982. Sediment ripples and dunes. Ann. Rev. Fluid Mech. 14, 13-37. Fredsoe, J., Deigaard, R., 1992. Mechanics of coastal sediment transport. Advanced Series on Ocean Engineering, Vol. 3. World Scientific, Singapore. Klemn R Rntunnn ~kamnmck_._, WC.-., 1997. Gn me r__r-D-‘_-_. nrnnnpatinn of _.., i____..-_ -_ internal bores. J. F!ujd M_ech. 331, __.__..r, ”1.-., __“__...._, R 81-108. Longuet-Higgins, M.S., Stewart, R.W., 1964. Radiation stress in water waves: a physical discussion with applications. Deep-Sea Res. 11, 529-562. Lott, D.F., Poeckert, R.M., 1996. Extending cooperative research. Canada, New Zealand, United States in joint effort in British Columbia to evaluate penetrometers for ground-truthing acoustic classifiers for mine countermeasures. Sea Technol. 9, 56-61. Mei, CC., 1982. The Applied Dynamics of Ocean Surface Waves. Wiley, New York. Mei, C.C., Liu, P.L., 1993. Surface waves and coastal dynamics. Ann. Rev. Fluid Mech. 25, 2-240. Neilsen, P., 1992. Coastal bottom boundary layer and sediment transport. World Scientific, London. Peregrina, D.H., 1983. Breaking waves on beaches. Ann. Rev. Fluid Mech. 15, 149-173. Sleath, J.F.A., 1984. Sea Bed Mechanics. Wiley-Interscience, New York. Voropayev, S.I., Roney, J., Fernando, H.J.S., Boyer, D.L., Houston, W.N., 1998. On the motion of large l.~+t~m Iro..+;r*,o ;nrl.,,..%i nrr;,,.,tCx,.., r ./vuaUU mort01 Pnn rP.,iPI., U”LL”LLl ~~nrlr0 :,I, .TlllliP **LL”ti-IIIU”&tiU “&.ULU’“‘J on.., IgUn. a. YL’&, .,nAnr ULIYQIL1”lln. White, F.M., 1991. Viscous Fluid Flow. McGraw-Hill, New York. Whitham, G.B., 1955. The effects of hydraulic resistance in the dam-break 399-407. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York. Yalin, M.S., 1977. Mechanics of Sea Bed Transport. Pergamon, NY.
problem. Proc. R. Sot., Ser. A 227,