NUMERICAL SIMULATION OF SCOUR AROUND FIXED AND SAGGINGPIPELINES USING A TWO-PHASE MODEL by - Zhihe Zhao A Dissertation Presented in Partial Fulfillment _ of the Requirements for the Degr
SIMULATION OF SCOUR AROUND A FIXED PIPELINE
Overview of Scour ModeÌs -cc cQ H nSnnnHY KH nh hy tên 2 1.4 Ho o:aiadaaẳiẳllẳẳaẳẦẳiẳiẳaẳảảọả
Mao [28] applied a modified potential flow theory and sediment continuity equation to simulate scour below long cylinders placed on a sandy bottom subject to a mean current.
The model predictions were compared with complementary laboratory experiments, and the potential flow model was able to simulate the upstream portion of the scour hole satisfactorily Li & Cheng [23] also developed a scour model based on the potential flow theory Instead of using an empirical sediment transport formula, they calculated the equilibrium scour pit size by assuming that the bottom shear stress everywhere on the seabed is equal to or less than the far field shear stress when the equilibrium state is reached A boundary adjustment technique based on Newton-Raphson method was utilized for simulations Their model also predicted the approximate upstream scour depth reasonably well, but failed to give correct predictions for the downstream part of the scour hole as a result of the limitations of potential flow theory Later, the same authors [24] solved flow equations by employing the Smagorinsky sub-grid scale (SGS) closure The predicted equilibrium scour hole agreed well with the experimental results.This approach, however, solely relied on the above-mentioned assumption on the bed shear stress, and it only produced equilibrium scour profiles, not the scour evolution.
Leeuwestein et al [22] used a k—£ turbulence model coupled with a sediment transport equation to simulate scour around pipes In this study, only the bed-load transport was considered first, and in this case the main sediment transport occurred by © ripples This prediction could not be corroborated by laboratory experiments Suspended sediments were then included in modeling, whence the ripples disappeared, indicating that bed-load transport alone is not sufficient for representing complex sediment transport processes around solid objects Brứrs [3] utilized a finite element method to solve the RANS equations with k-eé closure, while simultaneously solving a sediment transport model that included both the bed-load and suspended-load transports using a finite difference method The overall agreement between the predicted and measured scour evolution of Mao [28] was good However, the scour development almost stopped after
100 minutes into the simulation, although the measurements show continuous scouring even after 300 minutes [28] In addition, during the 6,000 bed profile updates conducted in Brers' simulations, problems were often encountered with regard to numerical instability of the bed update scheme as a result of the non-linear bed-load formula used. Liang et al [26] performed simulations of scouring around pipelines using a similar approach Two turbulence models, a standard k-¢ model and the Smagorinsky SGS model, were applied To avoid the appearance of unrealistically sharp irregular scour profiles and numerical instabilities during their calculations, a special smoothing technique, known as the sand-slide model, was employed in the simulation Simulations with the SGS model showed intense vortex shedding during scour, but scour profiles obtained with the k-é€ model were more realistic when compared with Mao's
4 measurements Therefore, Liang et al recommended the use of a standard &—e£ model for scour predictions.
Dupuis & Chopard [10] proposed a Lattice Boltzman Method to simulate scour around pipelines In this method, fictitious fluid and sediment particles moved on a regular lattice synchronously at discrete time steps, and time-dependent erosion processes involved were simulated However, only a portion of the equilibrium scour hole could be quantitatively compared well with laboratory measurements The model needed to be
“tuned” to match various other cases, thus limiting its utility as a robust predictive tool. Ali & Karim [1] used CFD software FLUENT to predict the three-dimensional flow and bed shear stress over a rigid bed By employing experimental data and the one- dimensional sediment continuity equation, they derived the variation of maximum scour depth with time as a function of the dimensionless bed-shear stress and streamwise distance Field measurements of scour around bridges were also applied to further verify the latter result Since the numerical simulation was only limited to a rigid bed, the scouring was not simulated explicitly.
1.3 OVERVIEW OF USING TWO-PHASE MODELS ON SEDIMENT TRANSPORT CALCULATIONS
In recent years, two-phase models, which consider the dynamics of particle and fluid phases as well as interactions thereof, have been employed for sediment transport calculations in the framework of Navier-Stokes equations Such models predict sediment transport from somewhat more fundamental (though modeled) dynamical equations, thereby avoiding the use of purely empirical sediment transport formulae Such formulae are replete in literature and have been found to be case dependent, thus limiting their general use to cover a broad range of flow configurations The two phase formulations, on the other hand, are developed based on more fundamental concepts, though naturally some parameterizations are required for closure As such, such models are expected to have more general applicability to a range of problems.
With regard to two-phase models, Yeganeh et al [48] used an Euler-Lagrange coupled two-phase model to simulate bed-load transport under high bottom shear Although the experimental results have shown the existence of a three-layer type velocity profile, the model produced only a two-layer velocity profile The authors ascribed this discrepancy to the neglect of inter-particle collisions in the model Hsu et al [18] employed a two- phase model to simulate suspended sediment transport, and demonstrated the ability of such models to predict the time-averaged concentration under a range of conditions.Greimann [16] employed a two-phase model to compute the average velocity of bed-load and suspended-load sediments in a laboratory flume under two-dimensional uniform flow conditions To calculate the coefficient of momentum loss a particle would experience when it is in contact with the bed, a critical Shields number was specified based on the particle shape and bed characteristics; the measured and calculated sediment velocities showed a reasonably good agreement In this study, the two-phase flow equations were used to calculate the velocity and concentration profiles of the sediment phase only.Greimann argued that there was no sufficient experimental data or analytical understanding of particle-turbulence interactions to develop a reliable two-phase model
6 for the flow phase Wanker et al [43] calculated sedimentation and sediment transport using an Euler-Euler coupled two-phase model The numerical model predicted the movement of a sand mound well, and they concluded that the bedforms are dependent mainly on the momentum exchange and particle-particle interaction terms.
In the present study, an Eulerian two-phase model embedded in FLUENT software is employed to simulate scour around pipelines The aim is to evaluate the model efficacy using available benchmark data and, if successful, to use the model to educe important information on flow dynamics, especially those that could not be conveniently obtained with available laboratory techniques The flow-particle interaction and particle-particle interactions are considered in the model formulation Each of the two phases (solid and fluid) is described using appropriately modified Navier-Stokes equations, and coupling between the phases is achieved through pressure and an interphasial exchange term For the solid phase, Boltzman's kinetic theory for dense gases is modified to account for the inelastic collisions between particles In order to include the effects of granular friction between particles for the cases of highly concentrated beds, the frictional viscosity derived from plastic potential theory is used [32] The simulation results are validated using experimental data available in literature.
It is well known that there is no general universally accepted formula to quantify sediment transport over a range of conditions The sediment transport rate is one of the most important characteristic for the two-phase flow motion Finding an expression for the transport rate has fascinated so many scientists, Consequently, since the first bed-load transport formula by Du Boys in eighteen century (especially in the past three decades), numerous transport formulae have been produced by various authors Among them, some representatives are E.Meyer-Peter, R Miiller, R.Bagnold, H.A Einstein and M.S.Yalin
Also, numerical results based on available sediment transport models tend to be sensitive to the selection of the sediment formula In the present simulations, however, there is no need for the selection of an empirical formula, given that such transport is handled using dynamical equations, underpinned to the extent possible by fundamental flow and sediment interaction mechanics The novel feature of our simulations is the use of two-phase flow theory to compute scour below a pipeline placed transverse to the flow Although two-phase flow theory has been applied for sediment transport calculations, it has not yet been applied for simulating scour In the latter case, the problem is more complex and due consideration should be given to the evolution of bed profiles.
MATHEMATICAL DISCRIPTION OF THE TWO-PHASE MODEL
Governing EQuatiOnS cóc HH HT KH nh kh rên 8 2.2 Turbulence Closure for Fluid Phase - cv seeee 10 2.3 Turbulence for Solid Phase .-.Q Q Qnn nh khu 12 2.4 Transport Equation for Granular Temperafure
The continuity equations for both the fluid f and solid s phases take the form: a(ứ,2,)+Ÿ*(œ,ứ,5,)=0, (1)ụ - where t=s,f and @ r+, =l;a r:ỡ,= volume fraction for water and sediment and
Py, P,= mass density of water and sediment, respectively.
The studies of the dynamics of a single particle in a fluid have identified the following important forces: the static pressure gradient; the solid pressure gradient, which is the normal force due to particle interactions; the drag force, caused by the velocity differences between the two phases; and the viscous force and the body forces Other forces, such as the virtual mass force and Basset force, are assumed negligible [4] The
` momentum equations for the fluid and solid phases, respectively, are [11]:
2(6/ỉ/1;)+V*(,príY,) =-a,VP+Vets+apprE+Ky(Ơ, -v,), (2)
= (a,p,5,)+V #(4,,5,5,) =~a,VP-VP, +Ve?, +ỉ,ỉ,8+K,ểŒ, ~v,), (3) in which ¥,, ¥,= the mean-flow velocity for flow and sediment; P = pressure shared by the two phases; 7; = stress tensor for the solid phase au, (Vi, + VV) +a, (A, ~3,)V-ÿ/1¡ Tụ = stress tensor for the fluid phase = ưru,(VY.+VY}) T= the identity tensor; 4, = bulk viscosity of the sediment =
(8)2 1 where b=(1+C,) (p,/p,+ c,y" and k,, is the covariance of the velocities of the fluid phase.
2.4 TRANSPORT EQUATION FOR GRANULAR TEMPERATURE
The granular temperature ©, for the solid phase describes the kinetic energy of random motions of sediment particles The transport equation derived from the kinetic theory takes the form [12]:
3| 02/z.,)+v bd (7.2.3,8.) = (-P,I+ Ts) : Vi, +Ve (ko, V9,) Yes + P (9) where (-P,J+?z) : Vỹ, = the generation of energy by the solid stress tensor, ky = the
: 15d JÐ diffusion coefficient = oP ONE TH + Le (4n -3)a,g,, + 16 41 —337)7@,2 1]>
4(41-337) 5 " 15z k, V8, = diffusive flux of granular energy, and =F +e, a
_ 1206) Bose p,a?@?, the collisional dissipation of energy, which represents the v 9, d, Vn 5 energy dissipation rate within the solid phase due to collisions between particles; óạ =-3K,©,, the transfer of the kinetic energy of random fluctuations in particle velocity from the solid phase s to the fluid phase ƒ#.
NUMERICAL SIMULATION AND VALIDATION: FIXED PIPELINES
Mao’s Experimental Set-up cuc HH HH HH nh Ha 14 3.2 Numerical ConfiguratiOn co HH HH HE ng eens xa 16 3.3 Simulation with Fluent con HH" HS nà nhe bà 18 4 RESULTS AND DISCUSSION: SIMULATION WITH A FIXED PIPELINE
A photograph of Mao's experiment is shown in Figure 2a and a schematic of the experiment is shown in Figure 2b A pipe with a diameter D = 0.1m was initially placed just above a sand layer of thickness 5, ~ 0.1 m (diameter of the sand particle d, = 0.36 mm); the sand layer depth was said to vary in 0.1 ~ 0.15 m, but careful examination shows that it is close to 0.1m (also see Section 4.2) A turbulent channel flow with Shields parameter 9 = 0.048 was introduced at time t = 0 Here the Shields parameter 6 is defined as ỉ = ———— , where 7 is the bed shear stress The pipe was held fixed sứ, - P,)4, by the end supports and the scour below the pipe was studied The channel was 2m wide,23m long with a height 0.5m The water depth was H, =0.35m (Fig 2b) The time variation of scour profiles was measured during Mao's experiments.
Velocity inlet Y Pressure outletelocity inle Water A xX
Figure 1 Numerical configuration for the simulation X is in the streamwise direction,
Y in the cross-stream direction and 6, the thickness of the sand layer.
(i) The Flume in Mao’s experiment.
In the numerical computations, the two-phase model described in Section 2 was set up to match the experimental configuration A logarithmic velocity profile with U,, =0.31 m/s was applied at the flow inlet The profile for k and Ê are given by referring to Brứrs[3].Pressure outlet boundary conditions, which require specification of gauge pressure at the outlet boundary, was applied at the flow exit The water surface is defined as the symmetry boundaries, wherein zero normal velocity and zero normal gradients of all variables are satisfied Wall boundary conditions were applied to the bottom of the sand layer.
In the simulations, a two-dimensional grid system with 9803 nodes and 9575 cells was generated with the grid generator GAMBIT of the FLUENT package The grid consisted of two zones, the water and the sediment A 105x60 non-uniform grid was mapped in the water zone with dimensions 2mx0.4m, and a 105x31 grid was mapped in the sediment zone with dimensions of 2mx0.1m (Fig 3) To match Mao’s experiment, the main simulations were performed with a sand-layer thickness of ở, = 1D, which constitutes the main results to be described in Section 4 Nevertheless, to document the possible influence of 6, , simulations were also performed with ở, = 1.5 D, the upper limit for 6, in Mao [28] The results of the latter are also presented, as appropriate.
18 The inlet and exit boundaries, respectively, were placed 5D and 15D (D being the diameter of the cylinder) from the center of the cylinder At the beginning of the simulations, a sinusoidal profile perturbation with amplitude 0.1D was introduced as a small disturbance to the initial bed profile (Fig 4).
FLUENT uses the segregate solver to solve equations (1) - (9) sequentially Firstly, the fluid properties are updated based on the current solution In order to update the velocity field, each of the momentum equations is solved using current values of pressure and mass fluxes at the faces A Poisson-type equation for the pressure correction is derived from the continuity and linearized momentum equations, which is then solved to obtain necessary corrections to the pressure and velocity fields in such a way that the continuity equation is satisfied Finally, the equations for turbulent quantities and granular temperature are solved using previously updated values of other variables.
As pointed out by Greimann [16], a bane of two-phase models is the inadequacy of the parameterization of particle-turbulence interaction terms; these terms are too strong and produce an unrealistically strong local reduction in the flow This, in our case, caused particles to settle rapidly, leading to an unrealistic pile-up of particles near the pipeline. Another issue is the time delay of flow adjustment following the scour When the bed profile varies, the flow needs time to adjust to the bed profile variation In the present simulations, this flow adjustment takes place on a time scale on the order of time that a fluid parcel takes to travel over the computation domain, which has a mismatch with the time scale where particle-turbulence interactions are taking place in the model This disparity of time scales can cause significant errors in scour calculations, aggravated by the fact that the response of flow to scour development is only approximately represented in two-phase model dynamics.
To avert the above problem, we opted to calculate the single-phase velocity field using the Navier-Stokes equations and k-¢€ closure (Launder & Spalding [21]) without taking into account the effects of particles The fully developed velocity field for the fluid phase so calculated (as a single-phase flow) was then used as the input field to conduct supervening two-phase model calculations (rather than obtaining the velocity field of the fluid phase using the two-phase model itself) Thereafter, the steady single-phase velocity field of the flow was calculated again with an updated bedform The procedure was repeated until the equilibrium state of scour was reached The interface between water and sand in the physical experiments was taken as that corresponding to the sediment volume fraction @, ~ 0.5 of the numerical experiments, as shown in Fig.5 (see Section
4.1 for justification) Figure 6 shows a typical example of a grid used in the flow calculations at an intermediate time, where 9567 nodes and 9300 cells are included; the bed profile shown corresponds to the contour level of a, ~0.5 obtained from the previous calculation step conducted using the two-phase model To obtain the fully- developed flow field, an adaptive grid that responds to the bedform evolution was used.During the simulations, the grid has been generated manually only once Thereafter, the grid was regenerated with the updated scour profile This was accomplished by running the journal file (a sequential list of geometry, mesh, zone, and tools commands executed during the first grid generation) using the grid generator GAMBIT, thus minimizing the total computation time. ơ
—~_ ——À + 0 ~ 3d,, Bed-load Layer l4 Laminated Load Layer (developing slowly)
Figure 5 A schematic diagram showing the interface, bed-load, suspended-load and laminated-load layers Here d, is the diameter of the sediment particles, H the depth of the water, Y, the level where the sediment volume fraction is at 0.5.
Figure 6 An example of the grid that was used for the flow model The bed profile is specified as the contour with a, =0.5obtained from the previous calculation step conducted with the two-phase model.
In this way, more realistic flow velocities could be maintained in the domain, thus alleviating rapid velocity profile changes characteristic of pure two-phase calculations.During the simulations, the time step size was chosen based on the number of iterations per time step [12], which was 30 - 40 to guarantee satisfactory results For the single phase flow model, the time step was on the order of 107's whereas the time step for the two-phase model was on the order of 10” s The flow model for a given scour state was run for a period of about three to four times the time it takes the flow to travel the computation domain to ensure that the flow is fully developed The corresponding duration for running the two-phase model was chosen so that the maximum change of scour depth along the sand-water interface is less than 0.03D for the first 10 mins and less than 0.01D thereafter.
Clear-water Scour SimulatiOn - co HH HH nh ki kệ 23 4.2 ‹ 09-0 BE 27 4.3 Sediment Transport MOd€S con HH HH nh kh 29
Figure 7 shows the results of bed profiles in the two-phase flow simulation described in
Section 3 As mentioned, the volume fraction of sediment contour a, ~ 0.5 was chosen as the bed profiles corresponding to the laboratory experiments in Mao [28] This selection was made in accordance with the experimental observations of Wang & Chien [42], which indicated the ‘laminar behavior’ of two-phase flows for a, 2.0.5 Note that our selection is consistent with the scour models in the literature [3, 26, 31], which have considered only the bed-load and suspended-load transports above the bed surface Our study, however, revealed that the sediments could still be in motion in the region below the bed-load layer although, in usual scour modeling literature, this region is assumed immobile Detailed measurements and analysis, however, have shown that the layer immediately below the bed-load layer can be in motion, leading to a “laminated load” [8, 42] This aspect is further addressed in Section 4.3.
The bed profile so determined computationally (Figure 7) is compared with Mao's experimental data in the right column; the agreement is very satisfactory Initially, the flow is subjected to blockage due to the existence of the transversal pipeline, and the flow beneath and above the pipe tends to accelerate Hence, the sediment particles underneath the pipeline have a tendency to be ejected fast The ejected sediments are supported by strong turbulent fluctuations, but further downstream, with the decay of turbulence (see Figure 8), the particles are deposited to form a mound As the scour depth continues to increase slowly at later times, the mound slowly moves away from the pipeline as a result of further downstream sand transport from the sand mound, above which the local flow
24 speed is larger (also see [41]) Finally, an equilibrium situation is achieved in such a way that particles flown into and carried out from the scour pit are in balance.
X{m) Figure 8 Normalized turbulent intensity at a location 1cm above the bed; w, is the particle settling velocity The inset shows the location of turbulence measurements.
The agreement between the predicted and measured scour shown on the right hand side of Figure 7 is highly encouraging, given the complexity of the model and the nascent nature of this work in simulating scour without invoking a purely empirical sediment transport formula The deviations in scour hole depth between the observations and predicted scour occurred at earlier times (t = 10 min), which could be attributed, at least in part, to the transient forcing of initially imposed sinusoidal disturbance.
The evolution of normalized turbulent intensity at a distance 1cm above the interface is shown in Figure 8 Because of the accelerating flow above and below the cylinder, initial turbulence levels therein are large, but with the development of scour, the flow velocity under the cylinder decreases, so do the turbulent velocity fluctuations Note that at large times the fluid turbulence intensity in the proximity of the scour pit approaches a value 4/1, ~ w,, where w, is the particle settling velocity, at a magnitude commensurate with that is necessary to keep particles in suspension This observation is consistent with the arguments of Stommel [37] and Boothroyd [5] that particles can be in a continuous state of suspension when the background turbulence velocities exceed the settling velocity [also see Noh & Fernando [30] and Biihler & Papantoniou [6]].
Figure 9 shows a plot of scour depth (i.e maximum depths of the scour hole as measured from the initial level) as a function of time Note the good agreement between the simulated scour depth and that measured by Mao [28] Also shown in the figure is the simulated depth with a sand layer of ở, =1.5D = 0.15 m, which shows a faster scouring rate during period between 10 min and 140 min, indicating that the bottom layer depth is a factor that determines the initial scour rate at least in the range of 6, = 0.1m ~ 0.15 m. The solid line shown is the commonly used scour prediction formula in literature [44]
5 =5,[1-exp(-2)], (10) where S, denotes the equilibrium depth and 7 is the time at which the scour depth reaches 63% of its equilibrium value The scouring rate calculated using a deeper sand layer agrees well with (10) over the entire time period Therefore, we infer that in Mao’s experiments the sand-layer thickness has been close to 1.0D Also note that equation (10) has been derived using measurements made with fairly thick sand layers, which may explain why the thicker sand layer showed a better agreement with it.
28 Also note that there is a slight oscillation of the scour depth about the equilibrium value at large times This has also been observed in the simulation of scour with Lattice- Boltzman method by Dupuis & Chopard [10] Sumer and Fredsge [38] compiled data from four previous investigations and suggested the average equilibrium scour depth S, under a fixed pipeline subjected to current as
The numerical simulation result of the present study gives S, /D ~ 0.6(Figure 9), which agrees well with experimental observations.
Scour depth(m) 0.05 + simulation with the thickness of a sand layer = 1.5D
@ Mao's measurement 7 © simulation with the thickness of a sand layer = 1D
Figure 9 The time evolution of scour depth in simulations and comparison with equation (10).
4.3.1 BED-LOAD, SUSPENDED-LOAD AND LAMINATED-LOAD
In practice, sediment transport is subdivided into several modes, and the common mechanisms often found in literature are the bed-load and suspended-load transports No precise definitions that help demarcate these modes clearly have been proposed thus far, although they represent two different mechanisms of sediment transport in the flow The bed-load is the part of the sediment load that is traveling immediately above the bed, supported by intergranular collisions rather than fluid turbulence According to Einstein [11], the bed-load layer is confined to a few grain diameters, within (2—5)đ, [45] On the other hand, the suspended sediment load is supported by fluid turbulence [13] These definitions, nonetheless, are rather qualitative and inadequate to describe complex dynamics of sediment transport near movable beds For example, when the shear stress is high, not only the particles at the interface but also those immediately beneath it start moving due to the penetration of momentum into the sediment layer by gradient transport and intergranular collisions [8] Unlike the bed and suspended loads, these sediments move in “laminar-like” layers, producing a laminated sediment load [8, 42].
Figure 10 illustrates the nature of sediment motion including the bed-load, suspended- load and laminated load at a river bed, based on Chien & Wan [8] On the same provisos,three sediment transport modes could be identified in the present simulations, as exemplified in Figure 11 For a better appreciation of these layers, the normal horizontal turbulence intensity profiles for t = 200 min are plotted in Figure 12, along different locations downstream of the cylinder It is clear that the turbulence dies off quickly below
30 the interface (a, 0.5), indicating that the drifting flow below the interface is of laminar nature According to Wang & Chien [42], when the volume fraction of sediment is greater than the threshold value of sand volume fraction, the particles are so closely packed that turbulence in the fluid is almost suppressed and the two-phase flow behaves as a laminar one, leading to the layers of laminated transport [8]. ằ Suspended load
Figure 10 Patterns of sediment motion from a flat bed, redrawn based on [8].
Figure 11 Vectors of the sediment velocity u/U,, at(a)t= 10 (b)t= 100 minutes
Figure I1 Vectors of the sediment velocity Z/, at (c) t= 200 minute.
Note that most of the numerical scour models in literature have considered only the bed-load and suspended-load transports For example, in Brers' simulations [3] discussed in Section 1, the sediment particles below the bed-load layer were assumed to be stationary and the results showed that bed development stopped completely after 200 mins, an observation that is at odds with laboratory results [28] The lack of laminated load may partly explain why the scour development stopped earlier in Brứrs' simulations vis-a-vis the experiments From Figure 11 it is clear that the laminated load, at least that in the layer immediately below the interface, plays a certain role in scour development.This observation suggests that frequently used sediment continuity equation [3, 26, 31] ought to incorporate the laminar load, in addition to bed and suspended loads, in scour calculations.
Figure 12 Normalized turbulence intensity profiles at various downstream locations (X = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6m) of the cylinder wake at t = 200 min. a, =0.5 is chosen as the bed profile (Fig.7) Note the scale for uy /w, on the upper left corner.
A noteworthy feature of the particle motion in Figure 11 is the presence of a recirculation zone beneath the scour pit and the adjacent sediment mound Analysis of computational output shows that this is a result the shear stresses near the water-sediment interface (which is transmitted downward through intergranular collisions and gradient transport, parameterized in terms of collisional viscosity, for example) and pressure gradients induced by flow surrounding the sand mound As shown in Figure 13, under the pipeline, the accelerating flow exacerbates interfacial stresses, which, aided by the
34 negative pressure gradient induced by the converging flow toward the sand mound is expected to cause enhanced laminated transport beneath the interface The decelerating flow downstream of the sand mound neither exerts a high interfacial stress nor does it produce a favorable pressure gradient for laminated flow As such, the laminated load that can be supported downstream of the sand mound is small To maintain the continuity, the flow under the sand mound forms a recirculating flow as shown in the
Figure 11, but it is spatially confined to the region where the driving forces are substantial. shear stress driven (aiding pressure gradient) shear stress driven Pipeline (opposing pressure gradient)
Figure 13 A schematic diagram for the formulation of recirculation in the sediment zone The sediment movement is driven by interfacial shearing force and pressure gradient.
Although the origin of laminated load can be explained as above, there are some issues related to the magnitude of sediment velocity below the bed surface Figure 11 displays vector plots of dimensionless sediment velocities The sediment velocity above the surface ranges from 0.7 to 1.3U), where U, is the background flow velocity, which is reasonable given the small response time (d 2 A 8y; ~ 0.007s, where „ = 10° m?/s is the kinematic viscosity of water ) of particles that allow them to approximately follow the fluid phase The sediment velocity of recirculation zone inside the sand layer (= 0.2U ,), however, is larger than that one would expect based on intuition, although there are no available measurements within the sand layer to corroborate our suspicion Perhaps this overprediction of sand velocity reflects the difficulty of modeling highly concentrated sediment flow as well as simplifying assumptions made in our simulations For example, frictional viscosity parameterizations employed could have yielded too low of a value, allowing excessive momentum diffusion below the interface Also, following previous works [3, 26, 31], it was assumed in the calculations that the scour profile does not vary during the flow adjustment (Section 3), which perhaps may not be tenable in reality albeit this assumption works well in scour profile calculations With sediment velocity higher than the normal, sediment particles in the recirculation zone (driven by velocity input from the single-phase flow simulation) may expedite the scour profile changes, thus compensating for the plausible reduction of scour velocity resulting from this assumption. These explanations, however, are speculative at best given that no relevant observational results exist on the laminated load Future studies should be directed at such studies.
4.3.3 CALCULATION OF BED-LOAD AND SUSPENDED-LOAD
Sediment loads in various layers were calculated as follows using the simulation results.First, the effective sediment-water interface, which separates the laminated-transport
Comparison with NBURY and DRAMBUIE Models
4.4.1 FORMULATION OF NBURY AND DRAMBUIE MODELS
At present, several models are in use for practical predictions of mine burial in the coastal zone, which includes Defense Research Agency Mine Burial Environmental model or
DRAMBUIE (developed by H.R.Walingford, U.K., 1994), NBURY (German Navy,
Stender [36], 1980) and Wave-Induced Spread Sheet Prediction or WISSP (U.S Navy,
1960's) Because of the operational convenience, these models use (sometimes overly) simple scour parameterizations mostly derived using field and laboratory observations [15, 41] Of these, DRAMBUIE and NBURY use scour formulae based on (limited) data collected in the presence of currents, and it is instructive to compare their scour predictions with the present numerical calculations NBURY implements the Carstens & Martin equation [15] derived using U-tube tests The sediment transport here is
1 characterized by the sediment Froude numberF =U,,[(s—l)gd, 1? U,, being the
(orbital) velocity above the boundary layer and a Froude number threshold for the mine burial is defined as F,=5.04(d,/D)'* For the present study, F = 4.126 and F, = 3.3719 For the case F > F,, the NBURY model calculates the maximum (scour) depth Y,, by solving the following equation,
0.01F + 7.07 where d, is the grain size, D the diameter of the cylinder, ứ the angle of repose of the sand and t the (tidal) current duration In the NBURY model, ifF > #,, the scour is assumed to occur by the suspended-load transport, which is consistent with the results shown in Fig.13 An alternative expression is used when F < F,.
4.4.2 COMPARISON WITH NBURY AND DRAMBUIE MODELS
Note that the NBURY formula does not incorporate a time period of current oscillations, and thus can be construed as applicable to steady currents of velocity U,,, although the model has been often used in the context of flow under waves with a maximum orbital velocity of U,, Although the main mechanism of mine burial in NBURY is assumed to be sand-ripple migration, it is instructive to investigate whether the main formula of this model is valid under conditions for which it was originally derived (i.e scour) Figure 15 shows a comparison of the present numerical model results for scour with those predicted by NBURY The latter shows a higher scour level compared to numerical results.
On the other hand, DRAMBUIE uses a current-induced scour formula based on observations around pilings, where
The time scale here is T, = 46D Play
= sediment density relative to water (2.65 for silicious sediment) and @ = Shields
, where A = 0.095, B = -2.02, g=9.81m/s’,s parameter as defined in Sec 3, which is related to the ambient flow away from the object.This empirical equation is a variant of (10), and its equilibrium scour depth is given by
U -0.75U S., =4S.mmx ————— #075U,