以水下ROV進行射流渫浚與自然河床沉載回填觀測之研究

行政院國家科學委員會專題研究計畫 成果報告

以水下 ROV 進行射流渫浚與自然河床沉載回填觀測之研究

研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 95-2221-E-002-105-

執 行 期 間 ： 95 年 08 月 01 日至 96 年 07 月 31 日

執 行 單 位 ： 國立臺灣大學水工試驗所

計 畫 主 持 人 ： 卡艾瑋

計畫參與人員： 研究生助理：彭子軒、粟群超、侯鈞耀、魏宛儀

處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 96 年 10 月 29 日

行政院國家科學委員會補助專題研究計畫

□期中進度報告

以水下 ROV 進行射流渫浚與自然河床沉載回填觀測之研究

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95-2221-E-002-105-執行期間： 95/08/01~96/07/31

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國立臺灣大學土木工程學系暨研究所

中 華 民 國 96 年 10 月 29 日

1

Jet trenching and backfill monitoring of river sediment beds

with an underwater ROV

1. Objectives and achievements

The overall research effort is aimed at developing a novel approach for the in situ

characterization of underwater sediment beds. The approach is based on the deployment of a

small-scale underwater Remotely Operated Vehicle (ROV) equipped with jetting and imaging

devices. To implement this approach, the project must address three challenges: 1) the

development and testing of a vehicle suitable for river deployment; 2) the development of

imaging techniques, to guide ROV navigation and measure the topography of the river bed; 3)

quantitative understanding of the jetting process, to be achieved through computational

modelling and laboratory experiments.

Key achievements that were documented in the previous report include:

1) The completion of the vehicle body and its tracked propulsion system, combined with

modelling of the resulting vehicle behavior over uneven terrain.

2) The development and testing of a laser-based strategy for simultaneous localization and

mapping (SLAM) over uneven terrain.

In this report, we document the following key achievements of the last year of the project:

3) The comprehensive development and laboratory validation of a theoretical and numerical

model of underwater jet trenching by plane jets. This is documented in detail by the first

journal paper provided in appendix:

Perng, A.T.H., and Capart, H. (2007) Underwater sand bed erosion and internal jump

formation by travelling plane jets. Journal of Fluid Mechanics, in press.

4) The extension of the approach to three-dimensional trenches created by point jets and more

complex swords with multiple jets. Partial results from this effort were presented at the 32nd

Congress of the International Association of Hydraulic Engineering and Research, in a paper

that is also provided in appendix:

Su, J.C.C., Perng, A.T.H., and Capart H. (2007) Underwater trench incision and turbid

overspill due to moving point jets. Proceedings of the 32nd IAHR Congress, Venice, Italy,

July 1-6, 2007.

2

powerful jets of underwater Remotely Operated Vehicles (ROVs) to bury submarine cables

and pipelines. Difficulties associated with these flows is that they involve a coupling between

fluid turbulence, sediment erosion and transport, and bed morphodynamics.

Figure 1. Underwater trench erosion and turbidity current induced by a moving transverse

plane jet, in a parallel walled channel, leading to an effectively 2D flow.

In this research, we have examined such flows using a combination of computational

modeling and laboratory experiments. To make progress in understanding and describing the

corresponding processes, we have first restricted our attention to simplified two-dimensional

configurations in which the flows are confined in the x-z plane.

3

Figure 1 shows the transverse plane jet configuration that we have investigated in order to

better understand the erosional action of moving underwater jets (Perng and Capart, 2007).

By choosing a plane jet and operating it within a parallel-walled flume, the flow can be made

uniform in the transverse direction, and can be modelled using a one-dimensional shallow

flow description, extended to account for erosion and entrainment.

Figure 2. More realistic trenching jets, mounted on an upright sword that moves into the sea

bed close to a transparent flume wall, yielding a 3D trench shape and flow pattern.

While such a simplification greatly facilitates analysis, it limits significantly the kind of

phenomena that can be treated. When dealing with flows that are of interest in practice, their

three-dimensional nature must be taken into consideration. The jetting configurations used in

practice by actual jet trenching vehicles, in particular, are not so simple (Capart et al., 2006).

4

resulting jet-induced flow and erosion patterns are three-dimensional, and should be treated

accordingly if more realistic results are desired.

The methods developed in the course of the study are described in greater detail below.

3. Experimental methods

Experiments are conducted at small scales in laboratory tanks of the type shown in Figure 3:

glass tanks that can be filled with water and beds of sand, subject to the entraining and

eroding action of moving high-pressure jets.

Figure 3. Laboratory tank used for the study of underwater jet trenching flows.

The water tanks rest on steel frames, and can be fitted with moving carriages and

high-pressure water pumps. Transparent walls on all sides are chosen to maximize optical

access, needed for both cameras and laser scanning devices.

5

Figure 4. Flow and imaging configuration for small-scale jet trenching tests. The jetting

device is a thin needle fed mounted on a moving arm of adjustable orientation and fed with

high-pressure water from an irrigation pump.

A typical configuration allowing three-dimensional jet trenching flows to be observed both

from the side and from the top is illustrated in Figure 4. In these experiments, the point-like

jet moves at constant speed along the sand bed, eroding and suspending sand that deposits

further away, backfilling the incised trench. Time-lapse images of a needle jet experiment,

imaged from two different angles with the use of a mirror, are presented in Figure 5.

7

4. Shallow flow modeling with erosion and entrainment

The modeling approach that we have developed for plane jets is a shallow flow description of

underwater jet-induced currents. The description takes into account turbulent entrainment of

ambient water as well as erosion and deposition of sand from the bed. The conceptual view

adopted for this model is illustrated in Fig. 6.

Figure 6. Definition sketch and conceptual view for jet-induced shallow eroding and

entraining bed currents.

The corresponding governing equations can be written

) 0 ( ) 0 (

cos

e

t

z

−

=

∂

∂

β

, (1)

₍

_h

_v

₎

_e

_e

s

t

h

−

=

∂

∂

+

∂

∂

, (2)

₍

_h

_v

₎

_e

_e

s

t

h

−

=

∂

∂

+

∂

∂

. (3)

)

(

)

(

h

v

i

i

s

h

t

∂

=

−

∂

+

∂

∂

_ρ

_ρ

(4)

₍

₎

_cos

_}

₍

₎

_sin

{

)

(

mv

g

h

h

g

j

j

s

mv

t

∂

+

−

−

−

=

−

∂

+

∂

∂

β

ρ

ρ

β

ρ

ρ

. (5)

and are based on conservation principles for mass and momentum, which must be

complemented by energy considerations (Perng and Capart, 2007). Sample results from 1D

computations are shown in Fig. 7 and compared with transverse plane experiments.

8

Figure 7. Sample results from our study of moving transverse jets: lines = computational

9

Challenges remain for the extension of such a modelling approach to three-dimensional jets.

The corresponding mathematical and computational developments will have to be pursued in

future work. Based on the experimental observations, however, a preliminary understanding

of the dominant processes has been obtained, and is illustrated in Figure 8. These observations

will help guide the choice of assumptions in future three-dimensional models.

Figure 8. Three-dimensional flow patterns induced along an erodible underwater sand bed by

slowly (top) and rapidly moving point jets (bottom).

5. References

Capart, H., Perng, A.T.H., Su, J.C.C, and others (2006) Small-scale laboratory trials of jet

trenching tools in sand. Research report, Hydrotech Research Institute, National Taiwan

University, November 2006, 166 p.

Fraccarollo, L., and Capart, H. (2002) Riemann wave description of erosional dam-break

flows. Journal of Fluid Mechanics 461, 183-228.

Fraccarollo, L., Capart, H., and Zech, Y. (2003) A Godunov method for the computation of

erosional shallow water transients. International Journal for Numerical Methods in Fluids

41(9), 951-976.

Perng, A.T.H., and Capart, H. (2007) Underwater sand bed erosion and internal jump

formation by travelling plane jets. Journal of Fluid Mechanics, in press.

formation by travelling plane jets

A. T. H. P E R N GA N D H. C A P A R T

Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taiwan

(Received15 October 2006 and in revised form 9 August 2007)

Theory and experiments are used to investigate the water and sediment motion induced along a sea bed by travelling plane jets. Steadily moving jets are considered, and represent an idealization of the tools mounted on ships and remotely operated vehicles (ROVs) for injection dredging and trenching. The jet-induced turbulent currents simultaneously suspend sand from the bed and entrain water from the ambient. To describe these processes, a shallow-flow theory is proposed in which the turbulent current is assumed stratified into sediment-laden and sediment-free sublayers. The equations are written in curvilinear coordinates attached to the co-evolving bed profile. A sharp interface description is then adopted to account rigorously for mass and momentum exchanges between the bed, current and ambient, including their effects on the balance of mechanical energy. Travelling-wave solutions are obtained, in which the jet-induced current scours a trench of permanent form in a frame of reference moving with the jetting tool. Depending on the operating parameters, it is found that the sediment-laden current may remain supercritical throughout the trench, or be forced to undergo an internal hydraulic jump. These predictions are confirmed by laboratory experiments. For flows with or without jump in which the current remains attached to the bed, bottom profiles computed by the theory compare favourably with imaging measurements.

1. Introduction

In estuarine, coastal and marine engineering, various underwater operations involve hydrodynamic action by tools that are moved steadily along the sea bed. Three examples are depicted in figure 1. Figure 1(a) shows a suction dredge, trailed behind a ship, used to extract sand from the bottom. For dense sands, jets may be mounted onto the suction head to loosen the soil prior to extraction (Zanker & Bonnington 1967). Another dredging technique which is entirely dependent on jetting action, called water injection dredging, involves trailing a multiport diffuser which discharges high-speed water onto the bed to entrain sediment into suspension (figure 1b). Provided that the sea bed is inclined, the suspended sand may then travel down the slope on its own in the form of a turbid plume (Knox, Krumholz & Clausner 1994).

A third task (figure 1c) has recently gained increasing importance owing to the development of offshore oil extraction and submarine transmission of power and data. The objective of this operation is to bury cables and pipelines under a protective layer of sand, of thickness of 1 or 2 m, in order to prevent damage by trailing anchors and fishing nets. Instead of performing the burial in two separate phases of excavation and backfilling, the operation is now often performed in a single pass using jet trencher

U U U (a) (b) (c)

Figure 1.Erosion of underwater sand beds by steadily advancing tools, moved along the sea bottom using ships or remotely operated vehicles (ROVs): (a) suction dredging, in which sand is mined from the bed; (b) water injection dredging (after Knox et al. 1994), in which jetting is used to induce a turbid plume; (c) cable burial by jet trenching (based on information from Fugro Engineers), allowing the sand bed to be scoured and backfilled in a single pass.

ROVs (remotely operated vehicles). The ‘product’, cable or pipeline, is first simply laid onto the sea bottom. The vehicle then uses two upright or oblique swords lowered on both sides of the product to inject high-speed water into the sand bed. The jets scour a temporary trench, travelling with the vehicle, which allows the cable or pipeline to descend into the sea bed before being buried under the re-depositing sand further downstream of the trencher. Similar tasks have also been carried out using the suction hoppers of dredging ships (figure 1a), with the pumps operated in reverse to produce jets instead of suction (van Melkebeek 2002).

Although extensive testing and operational experience has been accumulated for jet trenching vehicles (see for instance Machin 2001), the flow processes involved are not yet well understood. Practical difficulties hindering such understanding include operation in difficult and turbid environments, as well as complicated patterns of water and sand motions.

composed of uniform sand. By neglecting transverse variations and assuming that the flow pattern has attained a steady state in a frame of reference attached to the jetting tool, the general unsteady three-dimensional flow will be reduced to a two-dimensional steady flow that is more readily amenable to theoretical and experimental study. Because actual jet trenching operations involve powerful jets acting on beds of relatively fine material (fine to medium sand), turbulent suspension will be the only sediment pick-up and transport mechanism considered. To document such flows, we will first present laboratory experiments conducted with carriage-mounted jets in a tank of constant width. With guidance from the experimental observations, a shallow-flow theory of the jet trenching process will then be proposed. The predictions of the theory will finally be checked against the laboratory measurements.

The erosional action of water jets impinging on submerged sediment beds has been examined in a number of previous studies. Pioneering work was conducted by Rouse (1940), who carried out flume experiments with downwards plane jets and examined how the style and pace of scour-hole development varied with jet strength and sediment characteristics. Over recent decades, extensive experimental work on erosion by fixed jets has been performed by Rajaratnam and co-workers (Rajaratnam 1981; Aderibigbe & Rajaratnam 1996; Mazurek, Rajaratnam & Sego 2003), who examined plane and circular jets impinging at various angles of attack onto beds composed of different sediment materials. For jet-induced scour limited by threshold-of-motion effects, theories have been proposed by Hogg, Huppert & Dade (1997), Gioia & Bombardelli (2005) and Bombardelli & Gioia (2006), and tested against laboratory experiments. Other related experiments include those of Mohamed & McCorquodale (1992), Stein, Julien & Alonso (1993) and Hopfinger et al. (2004). In all these works, the eroding jets were kept at a fixed position relative to the sand bed. In the present work, by contrast, we document the special phenomena arising when the jets travel along the bed. First, it becomes possible for the scour hole to attain a steady shape (in a frame of reference moving with the jetting tool) despite ongoing net erosion and deposition of sand across the loose bed interface. Secondly, different speeds of advance yield distinct patterns of flow and scour, including shooting flows with and without separation, as well as flows featuring an internal hydraulic jump. Finally, provided that the speed of advance is not too slow, the jet-induced water and sediment motions become sufficiently well-behaved to permit development of a hydraulic theory.

To describe theoretically the jet-induced current and the resulting sand erosion, suspension and deposition, we will rely both on shallow-water theory (see e.g. Abbott 1979) and work on turbulent entrainment (see e.g. Ellison & Turner 1959). To merge these avenues together, we will follow the roadmap laid out in a seminal paper by Parker, Fukushima & Pantin (1986). Based on the energy approach inaugurated by Bagnold (1966), Parker et al. showed that the influence of turbulence and erosion on the dynamics of gravity-driven turbidity currents could be described by shallow-flow equations, provided that one keeps track of the energy budget in addition to mass and momentum balance. A similar approach has been adopted by Kobayashi and co-workers (Kobayashi & Johnson, 2001; Kobayashi & Tega, 2002) to model suspended sediment transport in the coastal surf zone.

To describe the geomorphic turbidity currents induced by travelling jets, we must make various refinements to this basic theoretical approach. First, we will introduce a

sublayered description in which the turbulent bottom current is assumed stratified into sediment-laden and sediment-free sublayers. A generalization of the sharp interface view adopted by Fraccarollo & Capart (2002) will then be used to treat erosion and entrainment across the interfaces bounding these sublayers. Secondly, the steep trenching fronts obtained near the point of impingement will motivate the adoption of curvilinear coordinates. In similar contexts, such curvilinear coordinates have been used in two recent works to describe turbulent entrainment by curved jets (Jirka 2006) and basal erosion by shallow subaerial landslides (Chen, Crosta & Lee 2006). Finally, we will transform the governing equations to moving coordinates in order to describe the steady flows observed after long travel times in a frame of reference attached to the moving jets.

The paper is structured as follows. Section 2 describes the laboratory experiments performed to characterize jet trenching flows. In§ 3, we derive our proposed sublayered shallow-flow equations. Section 4 is devoted to the construction of special travelling-wave solutions to these equations. The flow and solution details that must be addressed in order to compute long profiles of the bed and current are treated in§ 5. Comparisons between computations and measurements are presented in § 6. Finally, conclusions are proposed in§ 7.

2. Laboratory experiments

2.1. Experimental apparatus

To characterize the sand-bed response to the action of moving jets, small-scale experiments were performed at the Hydrotech Research Institute of National Taiwan University. The laboratory apparatus and procedure are shown in figure 2. Experiments take place in a rectangular tank having the following dimensions: length = 180 cm; height = 50 cm; inner width B = 12.6 cm. To obtain an unobstructed view, each sidewall is formed of a single glass panel of area 180 cm× 50 cm, with a thickness of 12 mm chosen to minimize deformation. The bottom of the tank is composed of an 18 cm thick layer of sand, submerged under a 29 cm deep clear-water ambient. Before each experiment, a scraping plate is used to give the sand surface a flat horizontal profile (dashed line in figure 2a). The level of the free surface of the water is kept constant during jetting by way of a siphon overflow. Photographs of the set-up are provided in Perng (2006).

The jetting device used to produce a plane jet is formed of three cylindrical jetting heads of width = 4.1 cm, placed side by side. Made of machined and welded copper, each head is supplied with high-pressure water (through a flexible polyurethane tube), and discharges the water at high speeds into the tank through a row of 19 nozzles of internal radius R = 0.25 mm. The resulting individual round turbulent jets can be expected to merge into a single uniform plane jet within distances of around 5 to 10 times the nozzle separation distance of 2 mm (Jirka 2006). The specific momentum flux per unit width of the corresponding equivalent plane jet is given by

Σ= hv2= Q

2

BNπR2, (1)

where N = 3× 19 = 57 is the number of nozzles, R their radius, B the channel width, and Q the total discharge fed to the jetting heads. The choice of three aligned but separate jetting heads is made to attain a more uniform spanwise distribution of jetting strength. In earlier tests, we found that we could not achieve the desired degree of uniformity with a single head spanning the entire tank width. The high-pressure

U U U B B Z α (b) (c) 29 cm 18 cm 180 cm

Figure 2. Experimental set-up: (a) side view of flume and moving jetting arm; (b) top view of flume and wide-angle camera used for time-lapse photography; (c) close-up of the travelling jetting heads and induced flow pattern.

water for the three heads comes from a three-way flow splitter, fed by a twin-piston irrigation pump. The pump is equipped with an air chamber to make the water supply very nearly steady, and its discharge can be adjusted to values up to Q = 125 ml s−1.

To move the jetting device along the bed, a motor-driven traverse is placed above the tank, and travels at a constant speed U that can be set in the range U = 1− 10 cm s−1. The jetting device can also be held stationary (U = 0) to observe the sand-bed response to a fixed plane jet. An articulated arm mounted rigidly on the carriage is used to position the jets above the sand surface. The connection between the arm and the jetting tool is a three-degree-of-freedom wrist used to adjust the jetting orientation. Precise adjustment is required to achieve a laterally symmetric erosion pattern. A slight spanwise asymmetry of the impinging jets can be amplified by the eroding flow into a tilted sand surface of up to 30◦ sideways inclination. For all the experiments presented below, the jet orientation was set to α = 60◦, and the

7 (a) (b) 6 5 Settling rate w (cm s –1) 4 3 2 1 0 0 0.1 Sediment concentration n [–] 0.2 0.3 z(1) z(0) w

Figure 3.Sedimentation column tests: (a) measured settling rates w(n) (filled circles) for various sediment concentrations n, compared with the fall speed W = w(0) of individual sand grains (open symbol, mean of 20 measurements); (b) definition sketch. Tests performed in water near 30◦C.

standoff distance of the nozzles above the undisturbed bed was set to Z = 5 mm. Three jetting discharges Q = 90, 107 and 122 ml s−1 were examined. This corresponds to water velocities at the nozzles of 8.0, 9.6 and 10.9 m s−1, respectively, and to jetting strengths Σ1= 5.8 l s−2, Σ2= 8.1 l s−2, and Σ3= 10.6 l s−2. For each discharge,

tests were performed at various speeds of advance U . Further tests conducted with another standoff distance Z = 10 mm are described in Perng (2006).

2.2. Sand properties

The sand material used for the tests is a medium quartz sand of density

ρS= 2670 kg m−3, median diameter d50= 0.33 mm, and coefficient of uniformity d60/d10= 2.0. The volumetric sand concentration in the static sedimented bed is estimated to be in the range 0.58 < n0<0.62. The settling of sand grains is known

to depend on the concentration of the suspension (Richardson & Zaki 1954; van Rijn 1984). Measurements of fall speed and settling rates were therefore conducted at various concentrations (figure 3). Fall speeds are measured by timing the fall of individual sand grains between two horizontal lines, whereas settling rates are measured using sedimentation column experiments (figure 3b). For each concentration

n, a known volume of sand is thoroughly mixed with water in a closed column, then left to settle under the action of gravity. Assuming that both the bed surface and turbid interface behave like kinematic shocks (see Ungarish 1993), their elevations z(0)

and z(1) _{will change at rates governed by}

dz(0) dt = nw n0− n, (2) dz(1) dt =−w, (3)

where w = w(n) is the settling rate at the given concentration. The settling process leaves both the suspended sediment concentration n and the total sediment volume

n0z(0)+ nh1 unchanged. In the actual sedimentation column tests, only the bed

interface z(0) _{is sufficiently sharp to be identified precisely. We therefore estimate}

the settling rate w(n) from the formula

w(n) =n0− n

n

dz(0)

Figure 4. Time-lapse images of a jet trenching experiment from start-up (top panel) until approximate convergence to a moving trench of permanent shape (bottom panel). Successive panels are separated by intervals of 2 s. Each black or white segment of the scale bar is 5 cm long. Conditions for this test are Σ = 8.1 l s−2 (medium jetting strength) and U = 6.8 cm s−1 (high speed of advance, leading to a shooting-flow pattern).

and time the rise of the sedimentation front dz(0)_{/dt either with a chronometer}

(for slower settling rates) or from time-lapse photographs (for higher sedimentation speeds). The resulting measurements are plotted in figure 3(a), with the mean fall speed of individual grains interpreted as the settling rate w(0) in the dilute limit n→ 0. The deduced settling speed decreases with concentration as a result of hindered settling effects. The observed settling speeds range from w(0)≈ 6 cm s−1 in the dilute limit to

w≈ 2 cm s−1 at concentration n = n0/2≈ 0.3.

2.3. Experimental procedure and observations

With reference to figures 2 and 4, moving jet experiments proceed as follows. The jetting tool is first positioned at one end of the tank, at the desired inclination and standoff distance above the horizontal bed. Carriage motion is then initiated, and shortly thereafter the pump is started. The ensuing bed erosion and sand suspension are observed through the tank sidewall. After travelling some distance along the bed, a stable flow pattern and scour-hole shape are established, and translate to the left with the jetting tool without significant further deformation. As required by conservation

(a)

(c)

(b)

(d )

Figure 5.Qualitatively different flow patterns observed when varying the speed of advance of the plane jet relative to the sediment bed: (a) fixed jet; (b) very slowly moving jet; (c) slowly moving jet; (d) rapidly moving jet. The direction of the jet-induced current is from left to right in all cases, and the jetting tool translates from right to left in (b)–(d).

of sediment mass (neglecting variations in sedimented sand concentration before and after trenching), the scour hole travelling to the left leaves behind a sand heap in the start-up region, with equivalent positive and negative volumes on either side of the initial horizontal bed profile. The experiment ends as the jetting tool reaches the opposite end of the tank, when both the carriage and pump are stopped.

The different flow patterns produced when varying the speed of advance of the jetting tool relative to the sand bed are shown in figure 5. Our discussion here is qualitative, but a phase diagram will be presented in § 6.2 to delineate these various regimes in a more quantitative manner. To provide a point of reference, the flow observed when the jetting tool is held stationary is first presented in figure 5(a). In that case, sand eroded from the bed forms two triangular heaps on both sides of a stationary scour hole, attaining a steady shape in less than 1 min. Within this leveed scour hole, a mixture of sand and water circulates counterclockwise around a spanwise vortical axis. Clear water injected at high speed by the jetting head churns around then discharges to the ambient by detraining at low speed across a boiling turbid interface separating the turbulent suspension below from the clear water above. The observed pattern is similar to that documented by Rajaratnam (1981) in his early stationary plane-jet experiments. Unlike the threshold-of-motion conditions considered by Hogg et al. (1997) and Gioia & Bombardelli (2005), here the bottom of the scour hole attains a dynamic equilibrium in which turbulent entrainment of sand from the bed is balanced by gravitational settling of sand grains out of suspension.

the jetting head), qualitatively different flow and scour patterns are then obtained depending on the speed of advance. For very slow advance speed (figure 5b), the jet-induced turbulent current separates from the bed after deflection by the scoured profile. After flowing along the curved bed in the form of a thin bottom layer, the sediment-laden jet fountains upwards, then rains down its suspended sediment. This pattern is similar to the strongly deflected regime observed by Rouse (1940) in his experiments on unsteady scour by fixed jets. For the present case in which the jetting device is moving, another interesting feature can be observed ahead of the point of impingement of the jet with the bed (figure 5b). Significant deformation of the bed occurs owing to the formation of a breaching front. The jet-steepened sand slope relaxes back towards the angle of repose, through shallow avalanching that extends for some distance in front of the jet. Breaching processes of this kind have been described by Van den Berg, Van Gelder & Mastbergen (2002) and Mastbergen & Van den Berg (2003). Even though the angle of repose is exceeded at the trenching front, this breaching response is not observed for faster speeds of advance (figures 5c, d), possibly because it is delayed by granular dilatancy and its required water infiltration from the ambient.

Figure 5(c) shows the pattern observed when the jetting head is moved slowly instead of very slowly along the bed. Close to jet impingement, the thin current flows along a steep trenching front where sand material is being continuously eroded from the bed. As it reaches the deepest portion of the travelling trench, the bottom current then thickens dramatically, undergoing an internal hydraulic jump. Upon expanding, the suspended sand layer adopts a thickness that exceeds the scour depth, and experiences a corresponding sudden slowdown of the longitudinal flow velocity. At the sudden expansion, a strong counterclockwise circulation is observed in the upper portion of the current, with a smaller clockwise circulation bubble observed near the bed. These are the hallmark features associated with a strong internal hydraulic jump. Downstream of this jump, the flow gradually quietens down, with the suspended sand settling back to the bed in a manner reminiscent of settling column observations, with distance from the jump replacing time elapsed as the independent variable governing the pace of re-sedimentation.

Finally, as shown in figure 5(d), a fast speed of advance produces a more elongated scour hole, in which the turbulent current forms a shooting flow that simultaneously entrains water from the ambient and interacts with the underlying sand bed. The current is erosional along about a quarter of the trench length, then depositional for the remaining three-quarters, until the suspended sand has settled out and the bed has recovered its original elevation. All along this course, the suspended sand current remains confined to a thin layer flowing rapidly along the bed. For the experiments shown in figures 5(b) and 5(d), in which bed visibility is good, it can be checked that the bottom elevation of the trench is close to uniform in the spanwise direction. The same is true for the flows of figures 5(a) and 5(c) when the jets are stopped and visibility is regained. Whereas this may not be the case in other scour problems (Bombardelli & Gioia, 2006), three-dimensional wall effects do not appear to influence the bed shape significantly in the present experiments.

For stationary jet conditions such as those of figure 5(a), it is clear that a prediction of the jet-induced vortical flow, entrainment and detrainment would require some rather elaborate computational modelling. As in the approach of Gioia & Bombardelli

(2005), probably the best that could be expected from a simplified theory would be to obtain basic scaling relations between the jetting strength and the dimensions of the scour hole, to be compared with empirical scaling relations such as those obtained by Rajaratnam (1981). Likewise, the complicated flow pattern of figure 5(b), where both breaching and flow separation are observed, appears to lie beyond the reach of a simple theory. The situation is more favourable, however, for the conditions of figures 5(c) and 5(d), where the flow appears sufficiently well-behaved to permit a hydraulic theory of the interaction between jet, ambient and sediment. In what follows, we restrict our attention to these two regimes, which also correspond to the conditions of practical interest in jet trenching applications.

2.4. Profile measurements

As shown in figure 6(a), time-lapse photographs like those shown in figure 4 are used to extract longitudinal flow profiles. The photographs are taken using a low-distortion wide-angle lens, capturing the entire tank length, and oriented with the image plane parallel to the tank walls. Image acquisition is performed at the rate of one image per second. To obtain quantitative measurements, profiles are extracted from these images using manual mouse clicks, then converted to physical coordinates using a calibrated scale factor and rotation adjustment. Coordinates in the travelling frame of reference are obtained by measuring distances with respect to a moving origin chosen as the position of the nozzle closest to the sidewall, projected vertically down onto the initial sediment bed profile.

As illustrated in figure 6(a), two profiles are extracted from each image. The first is the bed profile (filled circles), separating the motionless sand deposit below from the flowing current above. As for the settling column tests, this boundary is rather sharp and its visual identification does not suffer from much ambiguity. The second profile extracted (open circles) is an upper boundary of the zone occupied by suspended sediment. As seen in figures 4 and 6(a), this transition between sediment-laden and sediment-free regions is reasonably sharp in some places, but rather diffuse in others, leaving ample room for the subjectivity of the analyst. This second profile should therefore be taken as indicative only.

To gauge the rate of convergence of the trench profile towards a steady shape in the travelling frame of reference, the bed profiles corresponding to the time-lapse images of figure 4 are plotted together in figure 6(b). For clarity, the profiles are distorted vertically by a factor of 2, and plotted in the moving coordinates. They are observed to converge rather rapidly towards a steady-state limit, the more so for positions located closer to the jetting tool. Convergence is slowest for the far end of the trench, where bed levels exceed the initial bed elevation as a result of the start-up heap, which requires a significant travel distance to be left behind. With the exception of the distal tail, where profiles may not be fully converged, profiles captured near the end of a run are thus checked to approximate an asymptotic steady shape, and will be used in§ 6 for quantitative comparisons with steady calculations.

2.5. Simple model and characteristic parameters

Before developing a more complete theory, it is useful to sketch a simple model which explains how moving jets lead to the formation of a steady scour profile in a travelling frame of reference. Let z(0)_{, h and u denote, respectively, the bed height,}

(b) (c) –0.1 –0.05 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1.0 1.5 0 0.1 0.2 0.3 0.4 0.5 2 1 0 –1 –2 0.05 0 0 z (m) z (m) z– H xˆ/L xˆ (m) xˆ (m) t = 2 s 4 s 8 s 10 s 12 s 6 s

Figure 6. Longitudinal profiles: (a) bed profile (filled circles) and outer limit of suspended sediment (open circles) acquired manually from a digital image (time t = 12 s); (b) successive unsteady bed profiles (thin lines), converging towards a permanent shape approximated by the last profile (thick line) near the end of the experiment (vertical scale distorted by a factor of 2); (c) bottom profile (thick line) and current boundary (dash-dotted line) from a simplified model of the jet trenching process. Arrows indicate steady sand fluxes as perceived in a frame of reference moving with the jetting head.

trench. A simple unsteady equation for the local bed evolution can be written

∂z(0)

∂t ≈ −Eu + DW, (5)

where E is a dimensionless erosion coefficient, D is a dimensionless deposition coefficient, and W = w(0) is the fall speed in the dilute limit, all assumed constant for simplicity. At steady state in a frame of reference moving with the jetting tool (at

speed of advance U from right to left), the profile z(0)_{(x, t) reduces to z}(0)_{( ˆx) where} ˆ x= x + U t. (6) It follows that ∂z(0) ∂t = dz(0) d ˆx ∂ ˆx ∂t = U dz(0) d ˆx =−Eu + DW, (7)

which equates the flux of sand across the sloping bed surface to the rate of erosion and deposition by the jet-induced current. In figure 6(a), the thickness of the current is observed to vary roughly linearly with distance from the jetting head, i.e. h≈ C ˆx where C≈ 1/10. If we assume that the momentum flux of the current Σ = hu2 _is

approximately conserved as the current entrains bed material and ambient water, then u( ˆx)≈
Σ/ h( ˆx), and therefore

Udz (0) d ˆx =−E  Σ C ˆx + DW, (8)

which constitutes an idealized model of the trench response. The erosion rate is strongest near the jetting head and decays with distance, whereas the settling rate does not vary as long as suspended sand is available for deposition. The trench consequently deepens as long as erosion exceeds deposition, but rises back towards the undisturbed seabed level when settling starts to predominate. Neglecting the standoff distance between the jetting device and the sediment bed, this simple ODE can be solved immediately over the interval 06 ˆx 6 L to yield

z(0) H =−4  ˆ x L  1−  ˆ x L  , (9) where H= E 2 CD Σ U W, L= 4E2 CD2 Σ W2 (10)

are, respectively, the maximum depth and the length of the trench. The dimensionless profile obtained in this way is plotted in figure 6(c). The trench shape is asymmetric, with the point of maximum scour located at ˆx/L= 1/4, in qualitative agreement with the observed profile of figure 6(a). Approximate quantitative agreement for the trench depth and length for this and similar runs can be obtained by setting D = 0.6 and

E= 0.035.

The above simple model and approximate coefficient values can be used to provide rough estimates of various characteristic parameters. The aspect ratio of the trench is for instance H L = D 4 W U ≈ 0.15 W U, (11)

suggesting that trenches become reasonably shallow once the speed of advance U exceeds the fall velocity W . At the location of deepest scour, furthermore, the current velocity u and friction velocity u_∗ are given by

u1₄L=D EW, u∗ ₁ 4L  =√f u1₄L, (12)

where f≈ 0.05 is a bed friction factor (discussed further below). The corresponding ratio of friction velocity to fall velocity is then

u_∗ W =

D√f

Guy, Simons & Richardson 1966; figure 18 in van Rijn 1984). Suspension-dominated transport can therefore be assumed, as confirmed visually by the photographs of figure 5. We can also gauge the relative magnitudes of the water velocity, speed of advance of the jetting head, and particle fall velocity. For the present experiments, the current velocity at the point of maximum scour is approximately

u(1/4L) = DW/E≈ 1 m s−1. This is to be compared with speeds of advance U of the order of 5 cm s−1, and with water speeds through the jetting nozzles of the order of 10 m s−1. The velocity disturbance associated with the moving jetting head can therefore be safely neglected in comparison to the jet-induced water velocity. Likewise, the representative current velocity u = 1 m s−1 (at the point of maximum scour) can be compared with the fall velocity W = 6 cm s−1. As regards longitudinal flow motions, it therefore appears legitimate to treat the water–sediment mixture as a single phase (without interphase slip). For normal to bed motions, however, this is not the case, since the typical erosion speed (of the order of the jetting head speed at the trenching front) is of the same order as the fall velocity. Relative fluxes between the two phases must therefore be considered along the direction normal to the bed. Finally, for the test shown, maximum scour occurs approximately 10 cm away from the jetting head, or about 50 times the nozzle separation distance. While the round jets may not have merged completely before impinging with the bed, they can be expected to coalesce fully into a plane jet well before reaching the point of maximum scour.

These observations can be used to guide the formulation of a more complete theory, and overcome some key limitations of the simple model. It was assumed above that the coordinate ˆx is equivalent to the distance travelled by the current. This approximation is reasonable for the shallowest trenches (suchas the one shown in figure 6), but must be replaced by a curvilinear description for steeper trenching fronts such as those of figure 5(c) and 5(d). Most importantly, save for the settling of sand grains back to the bed, the simple model above does not account for gravitational influence. To deal with Froude-number effects (such as the formation of an internal hydraulic jump observed in figure 5c) and Richardson-number effects (the influence of density stratification on turbulence), a more elaborate theory is required and is developed in the next section.

3. Sublayered shallow-flow theory

3.1. Notations and assumptions

The various notations and assumptions used to derive the more complete theory are illustrated in figure 7. We consider the idealized situation sketched in figure 7(a). A plane jet submerged in a deep quiescent ambient impinges onto a loose stationary sediment bed, and we are interested in the ensuing pattern of water and sediment motion. Mean velocities are restricted to the x- and z-directions, and the flow is assumed uniform in the transverse direction. The jet-induced current takes the form of a turbulent bottom layer, flowing tangentially along the curved bed profile. Flow separation is not considered and the turbulent layer is assumed to remain attached to the stationary bed. Mass transfers occur both through the upper and lower boundaries of the turbulent current: quiescent water is entrained from the above ambient, and sand grains are eroded from the underlying bed. Sediment transport is taken to occur

z x s h1 h2 z(1) e(0) z(2) z(0) ρ0 τ₀(0) τ₁(0) τ₂(2) ι1 (1) ι1 (0) β ρ2 ρ1 ρ∞ ν ω ν –e(2) –e₍₁₎ p (a) (b) (c) (d)

Figure 7.Flow idealization: (a) jet-induced turbulent current, assumed attached to the curved bed profile; (b) suspended sand sublayer embedded within the turbulent current; (c) control volume with longitudinal fluxes and interfacial transfers; (d) mass drift and shear stress functions.

as suspended load, with eroded sand grains eventually falling back to the bed by gravitational settling.

To model the above processes, a sublayered shallow-flow description is adopted. Because the bed profile may adopt a steep slope near the jet impingement, boundary-fitted coordinates are used. The curvilinear coordinate s denotes arclength along the bed. The bed profile (x(0)_{, z}(0)_{) can then be integrated from}

∂x(0)

∂s = cos β, ∂z(0)

∂s =−sin β, (14)

where β is the local bed inclination below the horizontal. It is assumed that β is a slowly varying function of time and space, which can be considered constant on the scale of the adaptation time and length of the shallow current. Measured normal to the bed, the depth h of the flowing layer is considered small compared to the radius of curvature  of the bed profile, i.e. h  . The turbulent layer is further assumed to be sharply stratified into two distinct sublayers having depths h1 and h2, where h1 + h2= h. The lower sublayer of depth h1 is composed of a turbid

mixture of turbulent water and suspended sand. The upper sublayer, on the other hand, features turbulent water taken to be entirely sediment-free. These two sublayers are distinguished in order to let turbulent water detrain from the turbid layer. As suggested by the experimental photographs of figures 4 and 5, the overall jet-induced turbulent layer of depth h can be expected to expand monotonously owing to mixing with the ambient, yet the thickness of its internal turbid layer h1 can eventually

decrease owing to the settling of sand grains. In addition to the bed profile (x(0)_{, z}(0)_),

two other interfaces are therefore of interest: turbid interface (x(1)_{, z}(1)_{) marks the limit}

x(2)= x(0)+ (h1+ h2) sin β, z(2)= z(0)+ (h1+ h2) cos β. (16)

Here and throughout the developments below, subscripts are used to denote layer quantities (e.g. the sublayer depths h1, h2), whereas superscripts denote interface

quantities (e.g. the interface elevations z(1)_{, z}(2)_{). A definition sketch is provided in}

figure 7(b).

Denoting by n0 and n1the volume concentrations of sediment in the stationary bed

and turbid sublayer, the densities of the bed, sublayers and ambient are

ρ0= ρW+ n0(ρS− ρW), (17)

ρ1= ρW+ n1(ρS− ρW), (18)

ρ2= ρ_∞= ρW, (19)

where ρW and ρS are the mass densities of water and sand, and where subscript∞ is

used to denote properties of the ambient. In what follows, the bed sand concentration

n0and the densities ρ0, ρ2and ρ∞will be assumed constant, but the sand concentration n1and the resulting density ρ1 of the turbid sublayer will be allowed to evolve in time

and space.

Contrasting with the stationary bed and quiescent ambient, the jet-induced turbulent current has a high velocity v, shared by the turbid and clear-water sublayers, and assumed to be oriented parallel to the local inclination of the stationary bed. The x-and z-components of the turbulent-layer velocity are thus approximated by

(ux, uz) = (v cos β,−v sin β). (20)

Adopting a boundary-layer approximation, normal-to-bed components of the velocity will be disregarded in the balance of momentum and mechanical energy. This would not be valid for the stationary and very slowly moving jets of figures 5(a) and 5(b), where a two-dimensional roller and flow separation away from the bed are observed. For the slowly and rapidly moving jets of figures 5(c) and 5(d), however, the approximation appears reasonable since the flow direction is predominantly parallel to the bed.

3.2. Governing equations

As illustrated in figure 7(c), balance of volume, mass and momentum can be applied to control volumes wrapped around the whole turbulent layer or around each sublayer separately. Volume balance leads to the following continuity equations for the bed elevation z(0)_{, and the depths h}

1 and h2 of the two sublayers:

cos β∂z (0) ∂t =−e (0)_{, (21)} ∂h1 ∂t + ∂ ∂s(h1v) = e (0)_{− e}(1)_{, (22)} ∂h2 ∂t + ∂ ∂s(h2v) = e (1)_{− e}(2)_{. (23)}

In order to account for erosion and entrainment, terms e(a) _{on the right-hand sides}

respectively. Transfers are defined positive when volume is transferred from a lower layer to an upper layer. Conversely, transfers take negative values when volume goes from an upper layer to a lower layer, as expected for instance for turbulent entrainment from the outer ambient into the flowing current across interface z(2)_.

Because their mass densities remain constant, equations for the mass balance of the bed and sediment-free sublayers can be obtained simply by multiplying (21) and (23) by the corresponding densities ρ0 and ρ2. In the turbid layer, however, the sediment

concentration and associated sublayer density can evolve. An additional equation for mass balance is thus required and can be written

∂

∂t(ρ1h1) + ∂

∂s(ρ1h1v) = i

(0)_{− i}(1)_{, (24)}

where the terms i(a)_{, a = 0, 1, denote mass transfers across the corresponding interfaces.}

Again, mass transfers are defined positive when going up from a lower to an upper layer or sublayer. Two other useful equations can be obtained by linear combination of equations (21)–(24). The first is a balance equation for the overall mass m = ρ1h1+ρ2h2

of the flowing layer (per unit bed surface):

∂m ∂t +

∂

∂s(mv) = i

(0)_{− i}(2)_{, (25)}

where the mass flux i(2) _{across the outer interface is given by i}(2)_{= ρ2e}(2)_{. The second}

is a conservation equation for the total sediment mass

∂ ∂t{(ρ0− ρ∞)z (0)_{cos β + (ρ} 1− ρ∞)h1} + ∂ ∂s{(ρ1− ρ∞)h1v} = 0, (26)

where the first term denotes the local rate of change of the sediment mass contained in both the bed and turbid sublayer, and the second term is the divergence of the turbid flux. In this equation, there are no non-conservative products on the left-hand side, and no source or sink terms on the right-hand side. It is the only one of the governing equations that can be cast in such a pure conservation form.

Because the two turbulent sublayers are assumed to share the same velocity v, a single momentum equation is needed, with only along-bed momentum considered. The corresponding balance equation is obtained for a control volume enclosing both sublayers jointly (figure 7c). Neglecting non-hydrostatic effects, momentum balance in the normal-to-bed direction reduces to an expression for excess pressure (relative to a sediment-free hydrostatic water column)

p(y) = 

(ρ1− ρ∞)g cos β(h1− y), 0 6 y 6 h1,

0, h1< y, (27)

where y is a local normal-to-bed coordinate, p(y) is the local excess pressure, and g is the acceleration due to gravity. Including the thrust from these excess pressures, momentum balance in the along-bed direction s can be written

∂ ∂t(mv) + ∂ ∂s  mv2+1₂(ρ1− ρ∞)g cos β h21 − (ρ1− ρ∞)h1gsin β = j(0)− j(2). (28)

The terms on the left-hand side of this equation represent momentum change and flux, pressure thrust, and the along-bed component of the submerged weight. On the right-hand side, terms j(0)_{and j}(2)_{represent momentum transfers across the lower and}

upper boundaries separating the turbulent current from its underlying sediment bed and outer ambient. Upwards transfers are again defined positive. Equations (21)–(28)

(see e.g. Mathieu & Scott 2000).

3.3. Interface transfer relations

Let e(y) denote volume transfer across a line parallel to the bed at normal coordinate

y. We decompose the mass transfer i(y) across the same line as the sum of a convective transfer associated with e and a mass drift ι (iota) due to gravitational settling and correlated turbulent fluctuations:

i= ρ e + ι, (29)

where the mass drift can be written as Reynolds average ι =ρu_y, in which ρ and

u_y are local fluctuations in density and to-bed velocity. Likewise, the normal-to-bed momentum transfer j is composed of a convective transfer associated with mass flux i combined with a shear stress τ :

j= v i− τ, (30)

where the minus sign before τ reflects the usual convention in which a positive shear stress drives a downwards flux of momentum. Expressed as a Reynolds average of correlated fluctuations of the parallel and normal to bed velocities us, uy, the shear

stress can be written τ = −ρusuy for a homogenous fluid, with more complicated

triple correlations arising in the presence of density fluctuations.

Across the interfaces (x(a)_{, z}(a)_{) separating the flow sublayers from each other and}

from the bed and ambient, the quantities ρ, v, ι and τ may undergo discontinuous jumps. Notations ι(a)b and τ

(a)

b will be adopted to specify the mass drift and shear

stress on side b of interface a. Consider first the bed interface (x(0)_{, z}(0)_{). Turbulent}

suspension and gravitational settling drive a non-zero mass drift ι(0)₁ immediately above the interface on the side of sublayer 1. Underneath the interface, on the other hand, no mass drift occurs within the stationary sediment bed. Applied to an infinitely thin ‘pillbox’ control volume wrapped around the sharp interface, conservation of mass requires that the mass flux i(0) _{be the same on both sides, leading to the compatibility}

relation

i(0)= ρ0e(0)= ρ1e(0)+ ι(0)₁ , (31) where ρ0 and ρ1 are the mass densities on both sides of the bed interface. Likewise,

a turbulent shear stress τ₁(0) will be applied on the top side of the bed interface, to which the stationary sediment bed below can oppose a resisting shear stress τ₀(0). If the mass flux i(0) _{is non-zero, the two shear stresses will not be equal to each other.}

Instead, they must satisfy a second compatibility condition

j(0)=−τ₀(0)= vi(0)− τ₁(0). (32) Here no convective term appears on the lower side of the interface because the velocity in the stationary bed is equal to zero. The two compatibility relations (31) and (32) are analogous to the Rankine–Hugoniot shock relations governing hydraulic jumps and bores. They represent a generalization of the morphodynamic interface relation used by Fraccarollo & Capart (2002) to relate erosion rate and shear stresses at the base of an erosional dam-break wave. In this earlier work, no mass drift was considered.

For interface (x(1)_{, z}(1)_{) separating the turbid and clear-water sublayers, we have} i(1)= ρ1e(1)+ ι(1)1 = ρ2e(1), (33)

where mass drift is absent on the right-hand side because of the assumption that the top sublayer is entirely sediment-free. Because the two sublayers are assumed to share the same velocity v, no relation for the momentum flux j(1) _{is required. Finally, for}

interface (x(2)_{, z}(2)_{) separating the turbulent layer and outer ambient, compatibility of}

the momentum flux requires

j(2)= vi(2)_{− τ}(2)

2 = 0, (34)

where i(2)_{= ρ}

2e(2)= ρ∞e(2) and where τ2(2) is the turbulent shear stress applied below

the interface. There is no mass drift on either side because both the upper sublayer and outer ambient are composed of water alone. Furthermore, the right-hand side of (34) is zero because the outer ambient is assumed to be devoid of both current and turbulence. The mass drift and shear-stress functions discussed above are illustrated in figure 7(d).

The balance equations and interface relations can be combined to obtain two additional equations that will be useful for the calculations below. The first is an evolution equation for the mass density of the turbid sublayer

∂ρ1 ∂t + v ∂ρ1 ∂s = ι(0)₁ − ι(1)₁ h1 , (35)

which takes the form of an advection-source equation, where the source function is associated with mass drifts at the top and bottom of the sublayer. The second equation is an equation of motion for the turbulent layer:

∂v ∂t + v ∂v ∂s + (ρ1− ρ∞)gh1 m ∂z(1) ∂s + 1 2gcos β h 2 1 m ∂ρ1 ∂s = −τ(0) 1 + τ (2) 2 m , (36)

where the right-hand-side sink term involves two shear stresses: the free-shear turbulent shear stress τ₂(2) acting along the inner side of the upper boundary of the turbulent current, and the wall shear stress applied along the upper side of the erodible bottom τ₁(0) (see figure 7d).

3.4. Mass drift and shear stress functions

Up to this point, the balance equations (21)–(28) and compatibility relations (31)– (34) are direct consequences of our basic assumptions, combined with conservation principles. To complete the description, however, it is necessary to provide semi-empirical laws for the turbulent mass drift and shear stress functions. For the mass drift functions, we adapt the classical sediment suspension theory of Rouse (1937) and write

ι(0)₁ = ξ√k(ρ0− ρ1)− (ρ1− ρ∞) ω cos β, (37) ι(1)₁ = ξ√k(ρ1− ρ2)− (ρ1− ρ∞) ω cos β, (38)

where each drift function is expressed as the sum of an upwards turbulent diffusive flux and a downwards gravitational settling flux. The turbulent diffusive fluxes are assumed proportional to the square root of the specific turbulent kinetic energy k(per unit mass), multiplied by the density difference experienced across the corresponding interface. The non-dimensional proportionality constant ξ represents a ratio of eddy length l to shear-layer thickness δy, both assumed to scale with the depth h of the

concentration of suspended sediment n1 in the turbid layer, the effective settling rate ω will be lower than the fall speed of individual sand grains in unbounded fluid (Richardson & Zaki 1954). For the experiments described above, for instance, in which medium sand of diameter equal to 0.33 mm is used, the settling speed drops from 6 cm s−1 to 2 cm s−1 when the sand concentration goes from n1= 0 to n1= 0.3.

Within this range, the effective settling speed that best accounts for the observed trench response is found to be ω≈ 3 cm s−1.

For the turbulent shear stress τ₁(0) applied along the upper side of the stationary bed, we adopt the Ch´ezy-type formula,

τ₁(0)= ρ1f|v|v, (39)

applied by Kobayahsi & Johnson (2001) and Kobayahsi & Tega (2002) to the coastal surf and swash zones where fast-water currents induce high rates of suspended sediment transport. For non-dimensional friction coefficient f , Raubenheimer, Elgar & Guza (2004) obtained values in the range 0.02–0.06 for beach uprush and downrush (Dronkers 2005).The friction coefficient f is known to be influenced by various processes, including the particle Reynolds number (Bombardelli & Gioia 2006), turbulence modulation due to the presence of solid grains (Crowe 2000), and sediment transport intensity (Wilson 1989; Sumer et al. 1996). For the present jet trenching conditions, however, the manner in which these processes should be jointly parameterized is unclear, and we choose instead a constant friction factor f to be tuned by comparison with the experiments. Towards the higher end of the range quoted above, value f ≈ 0.05 is found to yield the best agreement and used for all the calculations reported below.

For the turbulent shear stress τ₂(2)applied on the inner side of the outer interface, the situation is close to free-shear turbulence, and we adapt the turbulent stress function used in standard one-equation turbulence models (Fredsøe & Deigaard 1992). By analogy with the Kolmogorov–Prandtl eddy-viscosity expression νT= l

√

k, the shear stress function is written

τ₂(2)= ρ2ξ√k(−v), (40) where −v is the velocity difference across the interface, and where non-dimensional parameter ξ is again the ratio of eddy length l to shear-layer thickness δy. Based on the assumption that the same turbulent eddies mix both momentum and density, the same constant ξ is used in (37), (38) and (40).

For the bed reaction shear stress τ₀(0), there is no need for any additional empirical function. Combining the two compatibility functions (31) and (32), we obtain

τ₀(0)= τ₁(0)− ρ0vι

(0) 1 ρ0− ρ1

. (41)

By Euler’s momentum theorem, momentum transport associated with mass flux across the assumed infinitely thin interface is balanced by a jump in shear stress. In the absence of mass flux, this jump vanishes and the identity τ₀(0)= τ₁(0) is retrieved in accordance with Newton’s third law. The resisting shear stress on the lower side of the bed interface is therefore fully determined once the mass drift and shear stress functions ι(0)₁ and τ₁(0)have been specified. We address in the next section the derivation

of an additional balance law for the evolution of the specific kinetic energy k of the tur-bulent velocity fluctuations, which intervenes in the three functions (37), (38) and (40).

3.5. Balance of mechanical energy

To close the description, energy balance must be invoked. The total mechanical energy of the system is composed of three different contributions: (i) the potential energy of the bed, current, and ambient system; (ii) the kinetic energy of the mean flow; (iii) the kinetic energy of the turbulent velocity fluctuations. For the latter, we do not distinguish between the solid and fluid phase, and consider the joint kinetic energy of the mixture. Approaches resolving the two phases separately have been proposed by Hsu, Jenkins & Liu (2003) and Ten Cate et al. (2004), but exceed the level of detail sought in the present work. Accordingly, the total energy flux across a vertical transect is due to advection of the different energy components complemented by the work of the excess pressure forces. The resulting balance equation is

∂ ∂t ₁ 2(ρ0− ρ∞)g z (0)2_{cos β + (ρ} 1− ρ∞)gh1  z(0)+1 2h1cos β  + 1 2mv 2_{+ mk +} ∂ ∂s ×(ρ1− ρ∞)gh1  z(0)+1 2h1cos β  + 1 2mv 2_+mkv+1 2(ρ1− ρ∞)g cos β h 2 1  v =−γ, (42) where the first term on the left-hand side is the time rate of change of the total mechanical energy (sum of potential energy, mean flow kinetic energy, and kinetic energy of the turbulent fluctuations, per unit bed surface), and the second term on the left-hand side is the divergence of the energy flux. Note that the potential energy is measured with respect to a ground state in which the settled sea bed rises to elevation

z(0)_{= 0 below a sediment-free water ambient. The symbol γ on the right-hand side}

of (42) is the energy dissipation function. Assuming an isothermal system, the second law of thermodynamics requires that this dissipation function be everywhere positive, i.e. γ> 0 (see e.g. Abbott 1979).

In order to obtain a balance equation for the kinetic energy of the fluctuations alone, we can form the quasi-linear combination

(equation (42))−1₂v2_{× (25) − mv × (36) −}_(ρ 0− ρ∞)gz(0)+ (ρ1− ρ∞)gh1/cos β × (21) −1 2(ρ1− ρ∞)g cos β h1− ρ∞g  z(0)+1 2cos β h1  × (22) − gz(0)+1 2h1cos β  × (24). (43) After repeated application of the chain rule, a considerable amount of algebra, and right-hand-side substitutions based on the compatibility relations (31)–(34), the resulting equation simplifies to

∂ ∂t(mk) + ∂ ∂s(mkv) = 1 2  τ₀(0)+ τ₁(0)v+ 1 2τ (2) 2 (−v) −12gcos β h1  ι(0)₁ + ι(1)₁ − γ, (44) which is one of the key results of the overall derivation. This equation is a balance law for the turbulence intensity k, where we recognize on the right-hand side the following source and sink terms. The first two terms are production terms associated with the work of the mean flow against the interfacial shear stress τb(a), and correspond to a

loss of kinetic energy by the mean flow. The third term is a sink term associated with the work of the turbulent fluctuations against gravity, and corresponds to a gain of potential energy induced by the upwards mass drifts ι(a)_b . Finally, the last term is the energy dissipation function, inherited from the overall balance of mechanical energy (42). It is seen from the right-hand side of (44) that both the production and