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International Journal of Multiphase Flow
journal homepage: www.elsevier.com/locate/ijmulflow
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The influence of relative fluid depth on initial bedform dynamics in closed, horizontal pipe flow
Hugh P. Rice a,∗, Michael Fairweather a, Timothy N. Hunter a, Jeffrey Peakall b,
a School of Chemical and Process Engineering, University of Leeds, Leeds LS2 9JT, UK
b School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK
a r t i c l e i n f o a b s t r a c t
Article history:
Received 14 September 2016
Revised 10 March 2017
Accepted 15 March 2017
Available online 24 March 2017
Measurements of time-dependent bedforms produced by the deposition of solid plastic particles in two- phase liquid-solid flows were performed using a novel ultrasonic echo method and via video image anal- ysis in a 100-liter, closed-pipe slurry flow loop. Results are presented for the settled bed thicknesses over a range of nominal flow rates and initial bed depths and are combined into several phase diagrams based on various combinations of parameters, with the bedforms categorized into five types. The novel observation is made that the type of bedform that arises depends on both the flow rate and the initial relative bed or fluid depth, with both ripples and dunes being observed in the same system and in a single experiment. In addition, the critical Shields number at incipient particle motion is measured to be θ sc = 0.094 ± 0.043, hysteretic behavior is observed, and the evolution and scaling of each time-dependent type of bedform is analyzed in detail and compared against several expressions for initial and equilibrium dimensions from the literature. A number of universal scalings for bedforms in any type of conduit are proposed with a view ultimately to unifying the observations of bedforms in pipes with those in channels and natural flows.
© 2017 Published by Elsevier Ltd.
Introduction
The hydraulic implications of moving and stationary parti- cle beds in low-velocity multiphase engineering flows have re- ceived relatively little research attention, despite considerable in- terest in understanding critical deposition and stratification in higher velocity flows (e.g. Rice et al., 2015a, b). The impor- tance of lower-velocity bedform dynamics is, however, also of critical concern, whether it be in terms of the enhancement or inhibition of solids transport, flow field properties, modifi- cation of effective surface roughness, conduit wear/corrosion, or the likelihood of solid plug formation and subsequent block- ages. In the case of nuclear engineering, for example, it is not known whether such flows are likely to affect the radiative en- vironment in terms of heat formation and radiochemical interac- tions with the conduit materials. In the case of civil engineer- ing, it is equally unclear whether bedforms may hinder the abil- ity of waste and run-off water systems to function efficiently∗ Corresponding author.E-mail address: h.p.rice@leeds.ac.uk (H.P. Rice).1 Current address: Faculty of Engineering, Architecture and Information Technol- ogy, University of Queensland, St Lucia QLD 4072, Australia(Banasiak and Tait, 2008; Banasiak and Verhoeven, 2008; Lange and Wichern, 2013; Matoušek, 2005; Skipworth et al., 1999).Despite the lack of research on mobile beds in engineering transport systems (either open or closed conduits), bedforms are ubiquitous in nature in both sub-aqueous (water-driven) and aeo- lian (air-driven) environments, and their laboratory analogs have been studied extensively by earth scientists in open and closed rectangular channels. As a result there is a large, mature canon of literature describing their behavior in natural and laboratory environments (see, for example, the reviews of Baas et al., 2016; Best, 2005; Charru et al., 2013; Coleman and Nikora, 2011; García, 2008). Bedforms can very broadly be grouped into three regimes: lower/sub-critical (lower plane beds, ripples, ripples on dunes, dunes), transitional (washed-out dunes) and upper or supercriti- cal (upper plane beds, antidunes, chutes and pools) (Graf, 1984; Simons and Richardson, 1961); here the focus is on ripples and dunes.In the case of open channel and environmental systems, rip- ples and dunes form at low flow rates, provided generally that particle diameter, d < 0.7 mm for ripples and d > 0.1 mm for dunes (Leeder, 2011, p. 132–137), all configurations of which prograde (i.e. progress downstream). Ripples form when the particle size and shear Reynolds number are both small (Raudkivi, 1997). Im-http://dx.doi.org/10.1016/j.ijmultiphaseflow.2017.03.007 0301-9322/© 2017 Published by Elsevier Ltd.No. runsportantly, their equilibrium sizes scale with grain size, but are independent of flow depth, whereas those of dunes scale with flow depth (Charru et al., 2013; Van Rijn, 1989). More specifically, van Rijn (1984c) also states that while both ripple heights and lengths are much smaller than the flow depth, dune lengths can be much greater than that depth.Particle density (103 kg m−3 )2.651.09, 2.652.651.38–1.432.67Unknown (sand) 2.742.41.242.652.490, 1.177, 1.0700.907Particle size (mm)0.091, 0.1470.05–0.5070.7101.15–1.902.00, 2.780.072.18, 3.060.1, 0.330.11, 0.870.132, 0.193, 0.5380.75660∼75671292232∼40n/a 112n/a n/aThere has been much debate over how the formation of rip- ples, dunes or other bedforms eventuate from a set of flow, particle and environmental conditions (e.g. Charru et al., 2013; Colombini and Stocchino, 2011; Fourrière et al., 2010). Most workers agree that ripples initially arise from a linear instability, however there is debate on whether dunes arise directly from a linear instability, or form through non-linear processes from progressive growth of ripples, referred to as pattern coarsening (e.g. Charru et al., 2013; Colombini and Stocchino, 2011; Fourrière et al., 2010; Kennedy, 1963; Richards, 1980). Here these arguments are briefly reviewed, starting with the stability analysis of bedforms that considers how an initial (sinusoidal) perturbation applied to a planar bed of par- ticles will be amplified or damped. In general, when a bed sur- face is perturbed, there are two possibilities according to whetherthe lag distance, δ, caused by the competition between the localMean flow velocity (m s−1 ) or Re8.93–1.500.12–0.370.69–2.030.72–2.270.56–6.500.507, 0.6761.2–1.6sediment transport rate and the velocity at the bed is positive or negative (Engelund and Fredsøe, 1982; García, 2008). If the bed is stable (δ > 0) then the perturbation is attenuated by changes in the flow and sediment transport rate, while if the bed is unsta- ble (δ < 0) the perturbation grows, resulting in wave-like bedforms (dunes, ripples, etc.).Re = 1800–12,000Re = 10−1 –104In a linear stability analysis of an erodible bed in the presence of a free water surface, Kennedy (1963) incorporated the lag dis- tance, δ, and categorized the resulting bedforms as dunes, plane0.30.263–0.703Re < 15,000beds or anti-dunes depending on the value of kδ and the flowConduitDimensions (mm)Ismail (1952) (in Acaroglu, 1968, p. 113)Thomas (1964)Wilson (1965) (in Acaroglu, 1968, p. 113)Kriegel and Brauer (1966) (in Acaroglu, 1968, p. 113)Acaroglu (1968)Nakagawa and Tsujimoto (1984) Takahashi et al. (1989)Kuru et al. (1995) Simkhis et al. (1999) Coleman et al. (2003) Ouriemi et al. (2009) Edelin et al. (2015)Rectangular Pipe Rectangular PipePipe Rectangular PipePipe PipeRectangular PipePipe27 × 7.625.1, 100.49.37 × 9.3753.576.2310 × 7549.731.140300 × 1003030depth, where k is the bedform wavenumber, but the model did not predict ripples. Richards later (1980) extended a model by Engelund (1970) and considered the effects of turbulence more thoroughly, finding two modes of growth corresponding to dunes and ripples. However, Charru and Mouilleron-Arnould (2002) note that these and other models tend to underpredict the wave- length of instabilities when compared to most experimental data. Colombini and Stocchino (2011) have further extended this ap- proach and conclude that both ripples and dunes do indeed arise as primary instabilities, rather than dunes forming as a result of “coarsening” of ripples. In contrast, Fourrière et al. (2010) and Charru et al. (2013) have argued that dunes do not arise directly from an instability, but rather by progressive growth (coarsen- ing) of ripples over time. Ripples have been observed to grow non-linearly after initial formation, through merger of bedforms since smaller bedforms travel faster than larger ones (Coleman and Melville, 1994; Fourrière et al., 2010).The engineering implications of bedforms in closed conduits areTable 1Studies of bedforms in closed conduits.Referencenot clear. It is not known, for example, whether time-varying bed- forms in pipes increase or decrease the likelihood of blockages. However, it is reasonable to assume that the morphology of bed- forms will influence the mean (and in turbulent flows, fluctuat- ing) velocity fields and particle flux strongly. It is surprising, then, how few studies of bedforms in equivalent phase spaces in closed- conduit systems exist (for example, Edelin et al., 2015; Kuru et al., 1995). The authors are aware of just 12 such studies, as summa- rized in Table 1, where the Reynolds number, Repipe , is defined as follows:RepipeUave D= ν , (1)where Uave is the mean axial flow velocity, D is the conduit diam- eter and ν is the kinematic viscosity of the fluid phase.Of the studies listed in Table 1, four were performed in rectangular channels and the rest in cylindrical pipes.Coleman et al. (2003) reviewed the majority of these studies and highlighted that available data on the generation and development of bedforms in closed-conduit flows are limited.Ouriemi et al. (2009) classified time-dependent bedforms ob- served in pipe flow as either “small”, “vortex” or “sinuous” dunes, the latter being observed only in turbulent flow. However, it is important to note that, in the sedimentology literature, “sinuous” refers to the latitudinal shape of dunes rather than the longitudinal shape, and the bedforms observed by Ouriemi et al. (2009) were strictly ripples, not dunes, based on their observed dimensions and the diameter of the particles used. They also presented a phasediagram of Re against Ga(H/d)2, where H is the fluid depth (i.e. H = D–h, where D is the pipe diameter and h is the bed depth), d is the particle diameter and Ga is the Galilei number, which isdefined as follows:2—d3 (s 1)gGa = ν2 = Rep, (2)where s is the specific gravity of the solid phase, g is the acceler-tions, it is clearly a pressing area of research. Here the initial, non- equilibrium behavior of time-dependent bedforms in two-phase, liquid-solid flow in a horizontal pipe is investigated using a novel ultrasonic backscatter method and via video image analysis. The overall aim is to derive a phase diagram of bedform types, for which several categories are defined, and the thresholds delineat- ing the types are identified.Background
It is important to identify some underlying properties of rip- ples and dunes, which are generally identified as different kinds of bedforms of “distinctly separate scales with no gradual transition” (Coleman and Nikora, 2011). García (2008) has reviewed some common ripple and dune scalings in terms of equilibrium height, wavelength and celerity (i.e. rate of progradation), and states that dune heights satisfy the following expression:hb 1≤
, (4)ation due to gravity, ν is the kinematic viscosity of the fluid and H 6Rep is the particle Reynolds number, defined as:—d[(s 1)gd]1/2Rep = ν . (3)In their phase diagram, Ouriemi et al. (2009) correctly predicted the threshold between a stationary and moving bed, and the de- velopment of “small dunes”, according to a linear stability analy-where hb is the bedform height and H is the fluid depth. However,µ d ¶this expression does not distinguish between dunes and ripples, as ripples are independent of flow depth and are smaller than dunes (Charru et al., 2013). Julien and Klaassen (1995) derived the fol- lowing relationship for dunes based on a compilation of laboratory and field data, which allows relative dune height to be related to particle size (d50):sis. Additionally, they found that the behavior of two categories of bedforms (“small” and “vortex”) were well separated on a plot of hb /D against Uavet/D, where hb is the bedform height and t is time.hbH = 2.50.350H. (5)In the majority of the studies listed in Table 1, only ripplesor dunes (and not both) were likely to have been encountered, based on the particle diameters used in the experiments: in theVan Rijn (1984a) derived a relationship that defined the criticaldune height ratio in terms of the critical shear flow velocities and particle size relationship as follows (van Rijn, 1984a):cases of the studies of Kriegel and Brauer (1966), Acaroglu (1968), Takahashi et al. (1989) and Simkhis et al. (1999), only dunes; in the cases of Ismail (1952), Nakagawa and Tsujimoto (1984), Wil-hbH = 0.110.3µ h ¶ ¡bH1 − e−0.5T¢(25 − T ), (6)son (1965) and Edelin et al. (2015) only ripples. Coleman et al., (2003) used two particle types of different sizes, one that was likely to produce ripples and the other to produce dunes. It is also important to note, as Coleman et al. (2003) have done, that measurements of the bed dimensions were taken while the flow was stopped in the studies of Nakagawa and Tsujimoto (1984) and Kuru et al. (1995). It is therefore possible that flow deceleration influenced the measurements in those cases.where T is referred to as the transport stage parameter, which is defined below:¡U r ¢ − (U )2 2=T τ τ,cr , (7)(Uτ,cr )2where Uτr and Uτ, cr are the particle bed shear velocity and the crit- ical bed shear velocity, respectively, and are as follows:There are several differences between open and closed, andcylindrical and rectangular conduits that would be expected to in- fluence the development and equilibrium dimensions of bedforms,Uτr =g1/2Uflow, (8)C’and therefore the structure of any phase diagram that describes them. In a conduit with a circular cross-section, the flow struc-2 ⎧⎪⎨⎪0.24(d∗ )−1 d∗ ≤ 40.14(d∗ )−0.64 4 < d∗ ≤ 10ture may be significantly modified by the presence of a thick bed (Adams et al., 2011); secondly, calculations of bulk quantities suchas mass flux must take account of the variation in chord length (i.e.bed width at the top of the bed) with height, although Kuru et al.(Uτ,cr )50⎪⎩(s − 1)gd= θs =⎪0.04(d∗ )−0.1010 < d∗≤ 20(9)≤0.24(d∗ )0.29 20 < d∗ 1500.055 d∗ > 150(1995), for example, in their study of erodible beds in pipes, as-sumed that the asymmetrical effects of the circular pipe are not significant when the shear layer is very thin compared to the pipe diameter. These differences aside, Coleman et al. (2003) havestated that, although the absence of a free surface causes bed-forms to develop more quickly in closed conduits, their equilib-where Uflow is the mean axial flow velocity in the flow area andd∗ is the dimensionless particle size and C’ is the particle Chézy coefficient, which are defined as (García, 2008; van Rijn, 1984a):³ ´d∗ = d50, (10)h (s − 1)g i1/3ν2rium dimensions appear to be the same as in equivalent open- channels flows. Thomas (1964) also noted that bedforms in open,closed, natural and artificial flow geometries appear to be closelyC’ =18log 4H , (11)d90related phenomena, and thus theories should account for both sys- tem types.Given the lack of data on bedform behavior in closed conduits and the engineering implications of the associated flow transi-and θs is the Shields number, a dimensionless shear stress definedas follows: τb sfθs = gd¡ρ − ρ ¢ , (12)where ρs and ρf are the solid and fluid densities, respectively, andτ b is the bed shear stress.Coleman and Eling (2000) and Coleman et al. (2003) found that the following expression for the wavelength, λb, of initial bedforms referred to as “wavelets” fit a large number of data taken from the literature:0.75λb = 175d , (13)with λb and d in millimeters, while various expressions for equilib- rium bedform wavelength have been given by Julien and Klaassen (1995), Raudkivi (1997) and Baas (1993, 1994) but are not repro- duced here for brevity, as the focus here is on initial bedform de- velopment.An important consideration is the choice of scaling param- eters for constructed phase diagrams. While the Shields num- ber is quite often used (see, for example, Edelin et al., 2015; Kuru et al., 1995) both Ouriemi et al. (2009) and Edelin et al. (2015) found that it was inadequate to describe ripple and dune transition in pipe flows and concluded that the geometry of the flow should also be incorporated. Alternatively, both Ouriemi et al. (2009) and Sumer and Bakioglu (1984) found that an instabil- ity occurred at a critical value of a characteristic Reynolds num- ber (a particle Reynolds number in the case of the former group, and a grain Reynolds number in the latter). However, Charru and Hinch (2006) delineated stable and unstable plane beds with a critical value of the Galilei number, Ga, which is defined in Eq. (2) and is a measure of the ratio of gravitational to viscous forces, of which gravity tends to be stabilizing and viscosity desta- bilizing. As measurements of the bed critical shear stress are much more difficult in closed conduits than in open channels, use of Ga as a scaling parameter is advantageous for pipe-flow studies. Addi- tionally, in any closed conduit, the absence of a free water surface means a no-slip condition exists at all boundaries, and anti-dunes and chutes/pools, the formation of which depends on the interac- tion between the bed and free water surfaces, cannot develop. The Froude number, upon which the delineation of sub- and supercrit- ical flows is based, is defined as follows:=Fr Uflow , (14)(gH )1/2where H is the fluid depth. Although other definitions of the Froude number exist (for example, Gillies et al., 2004), the one above permits comparison of open- and closed-duct, channel and pipe flows, since Uflow and H can be defined unambiguously in all cases.high-quality video footage of bedform development in a single ex- periment was recorded with a Nikon D3100 camera and a Nikon AF Micro-Nikkor 60 mm f2.8D lens. A variable centrifugal pump was used to control the flow rate, an impeller mixer to maintain a suspension in the mixing tank (nominal capacity 100 liters, i.e. 0.1 m3) and an electromagnetic flow meter to measure the flow rate, Q. The flow loop was filled with suspensions of the parti- cle species at several nominal (weighed) concentrations, φw, with φw = 0.1, 0.5, 1 and 3% by volume, and run over a range of flow rates, as described in more detail later (Section 3.3).Non-spherical plastic particles, the physical properties of which are summarized in Table 2, were used in all the experiments and were sourced from a blast media supplier (Guyson International, Ltd., UK). A micrograph of the particle species is given in the Sup- plementary Material. The particle size distribution was measured with a Malvern Instruments Mastersizer 3000 laser diffraction sizer and a Retsch Camsizer XT optical size and shape analyzer, and the density with a Micromeritics AccuPyc 1300 pycnometer.3.2. Distance calibration and bed depth measurementThe methods of positional calibration of the acoustic probe and bed depth measurement are as described by Rice et al. (2015a) with the same flow loop. The following parameters were used in the UVP-DUO system software, where r is the axial distance from the active face of the transducer: r0 and rmax are the min- imum and maximum measurement distances, respectively, with r0 = 5 mm and rmax = 50 mm; w is the width of each measurement bin and s is the separation between the central points of adja- cent measurement bins, with s = w = 0.37 mm. The flow radius in all runs was R = 42.6 mm.The acoustic transducer operates as an emitter-receiver, the root-mean-square (RMS) of the received voltage, V, as a function of distance from the probe being the basic measurement used to determine the bed depth: being solid surfaces, both (a) the lower pipe wall and (b) the top of a settled bed of particles are strongly reflective and produce a sharp peak in V. The position of (a) is used in a low-concentration test run to calibrate the exact position of the probe by finding its position relative to the pipe walls. The re- sulting positional offset, aoff, would then be applied to all subse-quent runs using this particular probe arrangement. The positionof (b) is then used to measure the total bed depth, h, over time in runs where a bed is present.The flow Reynolds number, Reflow , is defined as follows:In contrast to Yalin (1972), Kuru et al. (1995) found that “sand waves” form on initially smooth beds under laminar flow and thatReflowUflowH= ν , (15)turbulence is not required; similar observations were made by Coleman and Eling (2000) and Coleman et al. (2003). However, as described in more detail later, all ripples/dunes described here de- veloped under turbulent flow conditions.
Experimental method
- Particle and flow loop propertiesAn off-the-shelf ultrasonic system was chosen for the present study, consisting of a UVP-DUO signal processor (Met-Flow, Switzerland) and a monostatic (i.e. emitter-receiver) transducer op- erating at 4 MHz. The transducer was mounted, perpendicular to the mean flow direction, on a transparent, horizontal test sec- tion of a recirculating pipe flow loop with an inner diameter of D = 42.6 mm at a distance 75 D from the nearest fitting to ensure the flow was fully developed (i.e. statistically invariant in the axial direction) at the test section. More details of the flow loop em- ployed can be found elsewhere (Rice et al., 2015a). In addition,where H is the fluid depth, i.e. the distance from the lower pipe wall or sediment bed surface if present, to the upper pipe wall. The bed and flow geometry are as described by Rice et al. (2015a). For time-dependent bedforms, the total bed depth can be divided into two components such that:h = hb + hs (16)where hb is the bedform height and hs is the depth of the settled part of the bed, as illustrated in Fig. 1.3.3. Summary of experimental procedureThe experimental procedure for each set of runs was as follows:
- A known mass of solids corresponding to the nominal volume fraction, φw, was added to the mixing tank (φw = 0.1, 0.5, 1 and 3% by volume); the mixer and pump turned on to produce a fully suspended flow throughout the flow loop.
Table 2
Physical properties of particle species used in present study.Trade name Material Particle size, ith percentile, di (μm) Density, ρs (103 kg m−3 )
∗ Malvern Mastersizer 3000∗∗ Retsch Camsizer XT, based on equivalent-area diameterd10
d50
d90
Guyblast 40/60
Urea formaldehyde
269∗
468∗, 442∗∗
712∗
1.52
Fig. 1. Time-dependent bedform geometry and evolution. H and h are fluid and beddepths, with h = hb + hs , where hb and hs are bedform height and settled bed depth, respectively. R and D are pipe radius and diameter; ti is ith period between adjacent peak and trough.
- The pump rate was reduced to a low value for several (5–10) min, in order that suspended particles settled in the test sec- tion and a flat, stationary bed of particles that did not vary over time was generated. This was then the nominal initial bed depth for the run.
- The UVP-DUO (and video camera, when used) was set running, and the flow rate was set to its nominal value for the run.
- A known mass of solids corresponding to the nominal volume fraction, φw, was added to the mixing tank (φw = 0.1, 0.5, 1 and 3% by volume); the mixer and pump turned on to produce a fully suspended flow throughout the flow loop.
- Particle and flow loop propertiesAn off-the-shelf ultrasonic system was chosen for the present study, consisting of a UVP-DUO signal processor (Met-Flow, Switzerland) and a monostatic (i.e. emitter-receiver) transducer op- erating at 4 MHz. The transducer was mounted, perpendicular to the mean flow direction, on a transparent, horizontal test sec- tion of a recirculating pipe flow loop with an inner diameter of D = 42.6 mm at a distance 75 D from the nearest fitting to ensure the flow was fully developed (i.e. statistically invariant in the axial direction) at the test section. More details of the flow loop em- ployed can be found elsewhere (Rice et al., 2015a). In addition,where H is the fluid depth, i.e. the distance from the lower pipe wall or sediment bed surface if present, to the upper pipe wall. The bed and flow geometry are as described by Rice et al. (2015a). For time-dependent bedforms, the total bed depth can be divided into two components such that:h = hb + hs (16)where hb is the bedform height and hs is the depth of the settled part of the bed, as illustrated in Fig. 1.3.3. Summary of experimental procedureThe experimental procedure for each set of runs was as follows:
Results and discussion
- Threshold for incipient particle motionA total of four runs were identified – visually, then by inspec- tion of acoustic measurements for verification – as being at the threshold of particle motion. Using the method of Peysson et al. (2009) given in the appendix, a mean value for the critical Shields number – i.e. that at incipient particle motion (and see Eq. (12)) – of θsc = 0.094 ± 0.043 was measured, based on those four data and with the relative roughness constrained to ε/D∗ = 0.01 in each case (Peysson et al., 2009), where it is noted that ε/D∗ ∼ d/D; here ε is the roughness coefficient and D∗ is a length scale (see ap- pendix), and d50 /D = 0.011. The measured value of θ sc falls in the range given by Ouriemi et al. (2007) of θsc = 0.12 ± 0.03, based on a large number of data from the literature, and also agrees well with the direct numerical simulations results of Kidanemariam and Uhlmann (2014) in channel flow with several particle types and flow conditions.An estimate of the critical Shields number was also made using the method given by Soulsby (1997) – see the appendix – yielding a value of θsc = 0.038, i.e. a significant under-prediction of the mea-sured value given above. It is noted that the Soulsby (1997) methodwas employed by Edelin et al. (2015) and was found to accurately predict measured values in pipe flow with spherical, buoyant par- ticles (s = 0.910, d50 = 756 μm).
- Categorization of observed bedformsThe observed bedforms are categorized into five types accord- ing to their behavior and evolution over time (or lack thereof). Three examples of bedform fields that form from initially planar beds are shown in Figs. 2–4. In Fig. 2 (Q = 0.498 to 0.403 l s−1 at t = 0, φw = 0.1%) a ripple bed forms with a regular period, that grows to an equilibrium height, while in Fig. 3 (Q = 0.483 to 0.323 l s−1 at t = 0, φw = 0.5%) it appears that a ripple-bed forms ini-
Fig. 3. Bed thickness against time: transitioning ripple-to-dune formation with in- creasing period (Q = 0.483 to 0.323 l s−1 at t = 0, φw = 0.5%).
Fig. 4. Bed thickness against time: unstable and cyclic ripple-to-dune formation (Q = 0.383 to 0.287 l s−1 at t = 0, φw = 1%).tially (with a small frequency period) which transitions to a field of dune-type bedforms with longer period.The bed thickness in Fig. 4 (Q = 0.383 to 0.287 l s−1 at t = 0, φw = 1%) exhibits more complex behavior, varying strongly and cyclically, and the ripple/dune field is not at equilibrium at the endof the run (more than 30 minutes). The bed thickness exhibits bi- periodic behavior that suggests that at least two competing modes are present, and the bedforms quite abruptly transition from rip- ples to dunes and back again. Similar periodic behavior has been observed by Edelin et al. (2015) in studies of ripple formation with neutrally buoyant particles in pipe flow, although in their case, two distinct ripple formations were observed with differing wave- lengths and similar amplitudes, unlike in the present study.For the purpose of consolidation and summary, a list of ob- served bedform types is given in Table 3, categorized broadly and in order of decreasing flow rate. It is important to note that ripples of increasing period may, in fact, be of the unstable/cyclic type, as discussed in more detail below. However, while there are similari- ties between them, the differences are significant enough based on the evidence collected for two categories to be defined, although it is not known how universal the delineations are. For brevity, ex- ample runs are not shown for two further bedform types: upper plane beds and beds with no particle motion.In order to complement the acoustic data and verify the identification of bedform categories given above, a concatenated video from a single experiment (1200 seconds initial h/D = 0.278,Q = 0.191 l s−1, Reflow = 5341) was analyzed, the bed surface iden- tified and the wavelength, λb, and bedform celerity, cb (i.e. rate of progradation), computed, where the wavelength and celerity are related as follows:cb = fbλb. (17)Here fb is the bedform frequency and was calculated as the re- ciprocal of the bedform period at a fixed measurement point in the video images. The videos were recorded contiguously and were concatenated into a single dataset, as shown in Fig. 5. Ripples of regular period were observed, and mean values of the ripple frequency, wavelength and celerity of fb = 0.010 ± 0.003 s−1, λb = 75.2 ± 10.4 mm and cb = 0.731 ± 0.079 mm/s, respectively, were recorded, of which the wavelength compares well with the value predicted for the initial stage of bedform development – given in Eq. (13) herein, based on Coleman and Eling (2000) and Coleman et al. (2003), i.e. λb = 99 mm. The deviation in the appar- ent bed depth, δr, due to refraction in the pipe wall was calculated as a function of bed depth, h, but was not incorporated into the results shown in Fig. 5(c) because (a) δr was found to be small (0.2 r δr (mm) r 1 mm over the range 8 r h (mm) r 18), and (b) this optical distortion does not affect the measurements of bed- form frequency, wavelength and celerity, which are all calculated from extrema in the time series. Details of the calculation of δr are given in the Supplementary Material. It is also noted that optical distortion due to the lens itself was no more than 0.1% (DxO Labs). Of the bedform categories defined above, regular ripples corre- spond to a shorter-wavelength mode, which grows comparatively slowly (Fig. 2). The origin of the longer-wavelength mode (dunes) appears to be through one of two routes; via a progressive tran- sition from growing ripples (Fig. 3; and left hand-side of Fig. 4), akin to the ideas of pattern coarsening of ripples (Charru et al., 2013; Fourrière et al., 2010), or from a marked and rapid change in mode (right hand-side of Fig. 4) perhaps suggestive of an un- derlying instability. In the case of a progressive transition this pro- cess occurs quite rapidly (< 500–1000 seconds) perhaps reflecting the faster formation of bedforms in closed conduits than in open- channel flows (Coleman et al., 2003). The mechanism responsible for the unstable behavior observed in Fig. 4 at t ≈ 1000 s is not clear and has not been observed before, but it is suggested that it is a manifestation of competition between crest erosion at the highest parts of the bedforms, and modification of the flow area by bedforms: as bedform thickness increases, the effective flow area is reduced, and so the mean flow velocity and pressure drop increase,which acts to increase erosion and reduce bed thickness.Additionally, hysteresis was observed under a range of experi- mental conditions, of which Fig. 6 (Q = 0.483 to 0.323 to 0.483 l s−1 at t = 0, |∆Q| = 0.160 l s−1, φw = 0.5%) and Fig. 7 (Q = 0.342 to 0.277 to 0.342 l s−1 at t = 0, |∆Q| = 0.065 l s−1, φw = 3%) are two examples. In both figures, upon reduction of the flow rate (first frame of each figure) the initially planar beds develop an insta- bility in the form of ripples of increasing period transitioning to dunes (similar to Fig. 3). Upon increase of the flow rate again (sec- ond frame) the ripples are washed out and the bed tends towards a planar configuration: in Fig. 7 this occurs monotonically, whereas in Fig. 6 periodicity remains on the bed surface. This periodicity is transient and is likely to represent simple erosion of existing bed formations, the result of a competition between erosion and sed- imentation. The stabilizing effect that ultimately produces an up- per plane bed from ripples and dunes is thought to arise from in- creased sediment transport and therefore increased “crest erosion” (Charru, 2006). This mechanism has the effect of dragging parti- cles over the crests of bedforms – where the local fluid velocity is enhanced due to narrowing of the effective flow area – and into troughs (Andreotti and Claudin, 2013).
Table 3
Categories of bed/bedform identified in this study.Bed/bedform type Description- Upper plane bed Planar (i.e. flat) bed: depth does not vary over time, once equilibrated.Transitional between time-varying bedforms and heterogeneous suspension.
- Bedforms of regular period Ripples of time-independent period: period of ripples remains constant; depthmay increase with time. Example: Fig. 2.
- Bedforms of increasing period Ripples of time-dependent period which transition to dune formations; depthmay also increase with time. Example: Fig. 3.
- Bedforms with unstable/cyclic period Ripples and dunes of complex time-dependent periodicity, suggesting at leasttwo competing modes. Example: Fig. 4.
- Bed with no particle motion No particle motion occurs on surface of bed; determined visually. (Plots of suchruns show a flat line, so no example is provided.)
Fig. 5. Video image analysis results (Q = 0.191 l s−1 , φw = 0.5%). (a) Frame from video, example, image dimensions: 1920 by 1080 pixels, 75.7 by 42.6 mm; (b) binary (black and white) image of same frame as in (a), solid gray line indicates bed surface (flow from left to right); and (c) bed thickness, extracted from video, at two horizontalpositions (solid and dashed lines; solid upstream) separated by 27.6 mm, times corresponding to peaks and troughs (circles) also indicated for downstream position.
- Evolution and scaling of height and axial symmetry of time-dependent bedformsThe evolution of the height and axial symmetry of the three types of time-dependent bedforms identified in Section 4.2 (i.e. regular, increasing and cyclic) was investigated, with reference to a range of scalings from the literature described in Section 2. The three example runs are the same as those used in Section 4.2, the details of which are summarized in Table 4, and were cho- sen as being representative of their bedform type; hb and fb are the bedform height (see Fig. 1) and bedform asymmetry factor (see Eq. (18)), respectively.A MATLAB algorithm was used to smooth the bed depth data with respect to time and then identify the local minima and max-
Table 4
Summary of example runs to illustrate bedform evolution.
ima that correspond to the peaks and troughs of each bedform. The smoothing was necessary to eliminate scatter of the order of a few seconds (i.e. tens of samples), but was not found to affect the position or amplitude of peaks and troughs.The results of the algorithm are shown for bedforms of increas- ing period in Fig. 8, from which it is clear that the algorithm ef-Type
Flow rate, Q (l s−1 )
Nominal conc., φw
Regular
0.498 to 0.402
0.1
Increasing (ripples to dunes)
0.483 to 0.323
0.5
Unstable/cyclic
0.383 to 0.287
1
Fig. 6. Bed thickness against time from acoustic data, showing hysteresis behavior, (a) Q = 0.483 to 0.323 l s−1 at t = 0, (b) Q = 0.323 to 0.483 l s−1 at t = 0 (|∆Q| = 0.160 l s−1 , φw = 0.5%). Runs were performed concurrently.
Fig. 7. Bed thickness against time from acoustic data, showing hysteresis behavior, (a) Q = 0.342 to 0.277 l s−1 at t = 0, (b) Q = 0.277 to 0.342 l s−1 at t = 0 (|∆Q| = 0.065 l s−1 , φw = 3%). Runs were not performed concurrently.
Fig. 8. Total bed thickness, h, against time for case of bedform field of increasing period, with ripple-to-dune transition (Q = 0.483 to 0.323 l s−1 at t = 0, φw = 0.5%). Circles indicate local minima and maxima (i.e. peaks and troughs).ficiently identifies the peaks – which are signified by circles – of simple regular bedforms and bedforms of increasing period.The evolution and scaling of bedform height, hb, with respect to time was investigated, using the digitized profiles. Ouriemi et al.(2009) found that the relative height (hb /H) of different bedform types (specifically, “vortex” and “small dunes”) were very well de- lineated when plotted against a dimensionless measure of time (Uavet/D). Furthermore, several expressions are available in the sed- imentology literature for the equilibrium dimensions, hb /H, of bed- forms, as given in Section 2 (García, 2008; Julien and Klaassen, 1995; Ouriemi et al., 2009; van Rijn, 1984a, c). The calculated bed- form depths for each bedform type in those expressions were com- pared, at which point it should be noted that Uflow and the ini- tial value of H were used when calculating predictions of hb /H –and therefore C’, U∗r and T – according to Eqs. (4), (5) and (6). Thepredictions are overlaid on plots of the data as horizontal linesin frame (b) of each of the three figures below (i.e. Figs. 9–11). It is also important to note that the expressions against which the results are to be compared (specifically Eqs. (4), (5) and (6)) are strictly for predicting equilibrium bedform dimensions, whereas the bedforms in this study were not thought to be at equilibrium. Therefore, while there is a significant question as to the validity of these expressions in the present case, the purpose of making the comparison was initially to confirm that the observed bedforms were, indeed, not at equilibrium. Additionally, it allowed a qualita- tive judgement as to which expression for equilibrium dimensions was closest to the observed bedforms, had the experimental runs been longer.0.80.70.6hb/H0.50.40.30.20.10Present study Garcia (2008)Julien and Klaassen (1995) van Rijn (1984)The evolution of hb /H against t, is shown in Figs. 9–11, respec- tively, for each of the three bedform types considered above. The first clear observation to be made is that the bedform height is sig- nificantly smaller, with less scatter (with the exception of a small anomaly around t = 1000 s), in the case of regular ripples (Fig. 9) and appears to be at equilibrium by t = 500 s. The observed log- arithmic increase in ripple size over time is similar to that com- monly seen in channel flows (Malarkey et al., 2015) and is of a similar order to ripple formations found in floating beds of neu- trally buoyant particles in closed-pipe flow (Edelin et al., 2015).Importantly, the equilibrium profiles satisfy the definition of ripples, by being substantially under the hb /H ratio for dune transi- tion predicted by Julien and Klaassen (1995) and van Rijn (1984a),as given in Eqs. (5) and (6); hb /H also remains below the 1/6 value of García’s (2008) dune predictor in Eq. (4), although as0 500 1000 1500 2000Time, t (s)Fig. 9. Scaling and evolution of bedform height, hb , relative to fluid height, H, against time, t, for case of ripple bedform field of regular period (Q = 0.498 to 0.402 l s−1 at t = 0, φw = 0.1%).
Fig. 10. Scaling and evolution of bedform height, hb , relative to fluid height, H, against time, t, for case of bedform field of increasing period (Q = 0.483 to 0.323 l s−1 at t = 0, φw = 0.5%).
0.8Present studynoted earlier this expression does not distinguish between dunes and ripples. It is thought that these definitions, derived for open- channel flow, are however not valid for closed conduits such as pipes. Conceptually, the most important difference is the lack of a free surface: in a closed-conduit flow, the no-slip condition at the upper wall means the local flow velocity near the bed sur- face – and therefore the shear stress and, if the relevant conditions are met, the sediment transport rate – is likely to be larger than in an open-conduit flow with properties (i.e. cross-sectional area and bulk flow rate) that are otherwise the same. In this way, the discrepancy between the expressions of García (2008), Julien and Klaassen (1995) and van Rijn (1984a) and the bedform dimensions measured in this study in a closed conduit can be understood: the velocity and shear stress fields generate different bedform behavior than in open conduits.Inspection of Fig. 10 (bedforms of increasing period, show-ing a ripple-to-dune transition) and Fig. 11 (unstable/cyclic bed- form fields alternating over time between ripples and dunes) re- veals very similar initial behavior – a monotonic increase in bed- form height, albeit with some scatter, for 0 < t (s) < 500. At that point, however, the behavior of the two bedform types diverges: in the first (Fig. 10), hb continues to rise with a secondary phase, from t > 500, and does not reach an equilibrium height within the timescale of the experiment. In Fig. 15, the hb ratio becomes extremely unstable, with phases that appear at a ratio less than0.2 and more than 0.3, although with significant scatter. While quantitative distinctions between these oscillating cases are dif- ficult to make, there appears to be value in comparison to the dune transition correlations of Julien and Klaassen (1995) and van Rijn (1984a). hb /H ratios in both Figs. 10 and 11 approachthat for dune transitions as described by van Rijn (1984a), al-0.70.6hb/H0.50.40.30.20.10Garcia (2008)Julien and Klaassen (1995) van Rijn (1984)though the ratios are significantly below that of Julien and Klaassen (1995). The likely reason for the closer correlation to the van Rijn (1984a) relationship is that it contains a larger number of flow-specific parameters that more accurately account for pipe flow but which are absent from the García (2008) and Julien and Klaassen (1995) expressions.To provide further evidence for the ripple-to-dune transition, the axial symmetry of the three time-dependent types of bedforms was investigated, and was quantified by the bedform asymmetry factor, fb , defined as the ratio of the periods between adjacent min-ima and maxima in the bed depth, as illustrated by ti and ti +1 inFig. 1 and shown below:0 500 1000 1500Time, t (s)Fig. 11. Scaling and evolution of bedform height, hb , relative fluid height, H, against time, t, for case of unstable/cyclic bedform field alternating between ripples and dunes (Q = 0.383 to 0.287 l s−1 at t = 0, φw = 1%).t12fb = t . (18)Here, t1 and t2 are the larger and smaller of ti and ti +1 , respec- tively (which are illustrated in Fig. 1). It is clear, then, that fb was contrived to be the ratio of the larger to the smaller of adjacent periods, so that fb ≥ 1, in order to allow a clearer illustration of the development of asymmetry with time, where it would be expected
Fig. 12. (a) Bedform asymmetry factor, fb , against time for regular ripples (solidblack line; Q = 0.498 to 0.402 l s−1 at t = 0, φw = 0.1%), ripples with increasing pe- riod (dashed black line; Q = 0.483 to 0.323 l s−1 at t = 0, φw = 0.5%) and unsta- ble/cyclic bedforms (solid gray line; Q = 0.383 to 0.287 l s−1 at t = 0, φw = 1%); (b) schematic summary of bedform types.that formation of dunes would correlate to an increase in fb . Celer- ity and wavelength could not be measured with the acoustic sys- tem, as a single probe was used to measure bed depth. However, fb , although a ratio of periods, is intended to be a proxy for the ratio of wavelengths of the same bedforms, which is reasonable if it assumed that the change in celerity between adjacent bedforms is small.Plots of the bedform asymmetry factor, fb , are shown for the three example runs (see Table 4) in Fig. 12(a) for regular ripples, ripples with increasing period and unstable/cyclic bedform fields.In the case of regular ripples, fb remains small (fb < 2 for the vast majority of bedforms) and shows no significant trend over time, and comparison of Fig. 12(a) with Fig. 9 (hb /H against time) con-firms that the bedforms remain stable, quite axially symmetricaland of small amplitude relative to the pipe diameter and fluid depth. The trends for increasing-period ripples and unstable/cyclic bedforms are: an increase in asymmetry over the first few hundred seconds, followed by significant scatter; and much higher values of fb compared to the first case (regular ripples). When the evolu- tion of bedform height, hb (Figs 10 and 11) is compared with that of bedform asymmetry, there is a broad correlation that deeper bedforms are more axially asymmetrical, as would be expected for dune-type bedforms, with unstable/cyclic bedform fields becoming most asymmetrical. The bedform types are summarized schemati- cally in Fig. 12(b).
- Influence of solids volume fraction and particle sphericity and roundnessIn the first instance, the main influence of the solids volume fraction, φw – which is that of the whole system, as was the case in the study of Edelin et al. (2015) – is on the range of achievable initial bed depths. Specifically, higher volume fractions allowed for thicker beds to form, as the test section of the flow loop acted as a sink in which particles could readily settle. The values of h/D corre- sponding to φw = 0.1, 0.5, 1 and 3% by volume are approximately: 0 < h/D < 0.08, 0.08 < h/D < 0.18, 0.18 < h/D < 0.4 and h/D > 0.4, re-spectively.Gore and Crowe (1989, 1991) found that turbulence intensity in multiphase pipe and jet flows was either attenuated or en- hanced relative to equivalent single-phase (i.e. unladen) flow ac- cording to the value of d/le, where d is the particle diameter and le is the length scale of the most energetic turbulent eddies such that le ≈ 0.1 D (Hinze, 1959). Above and below d/le ≈ 0.1, turbulence in- tensity is enhanced or attenuated, respectively. In the experiments presented here, d/le = 0.110, and so the modulation effect is likely to be negligible.Once a bed had formed in the flow apparatus, the ambient vol- ume fraction appeared to be very low, as it was depleted by the existing bed, as is clear from Fig. 13 (same run as in Fig. 5: see also associated text), and from visual inspection during the runs. The similarity with images of bedforms in motion presented by Gao (2008) and others (Edelin et al., 2015; McLean et al., 1994; Raudkivi, 1963) is noted. Although in the present experiments the turbulence modulation effect was likely negligible, in general it would be expected to increase with solids concentration and affect the deposition behavior.The particle sphericity and roundness were measured in or- der to quantify their effect on settling velocity as it pertains to incipient motion and bedform behavior. The expression given by Dietrich (1982) for settling velocity – see the appendix – was eval- uated using measured values of the Corey shape factor, Fs (a mea- sure of sphericity) and the Powers roundness factor, P, measured with a Retsch Camsizer XT optical shape analyzer. The calculated settling velocities were then compared to those for equivalent par- ticles of perfect smoothness and sphericity to quantify the effects of those properties. The results are summarized below.
- The median particle size (i.e. d50) measured with the optical instrument and assumed to be that of a circle of equivalent projected area, was 442 μm, very close to the value of 468 μm measured with the Mastersizer laser-diffraction instrument (see Table 2);
- The Corey shape factor was measured to be Fs = 0.842 (where avalue of 0 corresponds to a rod and 1 to a sphere; Corey, 1949; Dietrich, 1982) with a corresponding reduction in dimension- less settling velocity, w∗, to 0.79 of that for an equivalent spher-ical particle (i.e. with Fs = 1);
- The Powers roundness factor was measured to be P = 2.49, cor- responding to a “sub-angular” shape (on a scale from 0: “veryangular” to 6: “well rounded”; Powers, 1953; Syvitski, 2007) with a corresponding reduction in dimensionless settling veloc- ity, w∗, to 0.91 of that for an equivalent well-rounded particle (i.e. with P = 6).
- The effect of both Corey shape factor and Powers roundness factor was to reduce the settling velocity, w∗, to 0.77 of that for an equivalent spherical, well rounded particle (i.e. with Fs = 1 and P = 6), where it is noted that the effects of Fs and P given above individually do not combine in a simple way.
Fig. 13. Images of flow over regular ripple bed (Q = 0.191 l s−1 , φw = 0.5%; 1920 by 1080 pixels, 75.7 by 42.6 mm). Dashed lines represent boundary between stationary and moving parts of bed, from inspection of detail in images. Images (a) and (b) from same run, separated in time by ∆t ≈ 3.5 minutes and showing different ripples. Circledareas show examples of common bed features in stationary part of bed.eroded and migrate more quickly than they would with equiva- lent spherical particles. Critically, the settling velocity is intimately involved in resuspension and incipient particle motion, which pro- ceeds when the bed shear velocity becomes comparable to the set- tling velocity (Hinze, 1959; van Rijn, 1984b). So, to first order, par- ticle shape might be expected to reduce the flow rate necessary for incipient particle motion, which is consistent with the value of the critical Shields number calculated here being lower than that predicted by Ouriemi et al. (2007), as described earlier.Lastly, it is noted that Clark et al. (2015) found that the thresh- old for particle motion has a strongly hysteretic character: that is, the threshold differs depending on whether the flow rate is de- creased until the particles cease to move, or if the flow is gradu- ally increased until particles on the surface of an initially station- ary bed begin to move. It is clear that this mechanism, along with the others described above, contributes to the complexity of bed- form behavior, that complexity being greater in the case of angular, non-spherical particles.
- Phase diagrams of bedforms in closed pipe flowA phase diagram of bedform types – as categorized in Table 3 – is presented in Fig. 14 in terms of the bulk Reynolds number, Repipe (Eq. (1)), against Ga(H/d)2 (see Eq. (2) for the Galilei number, Ga),where H and d are the fluid depth and particle diameter, respec- tively. It is noted that Ga is constant for a given particle species and so the quantity that varies in Fig. 14 is the ratio H/d. These variables are as used by Ouriemi et al. (2009) with silica particles. It should be noted that H is the initial value before the bed sur- face is perturbed by a change in flow rate. These quantities were chosen because they have a common interpretation in all the runs and could be evaluated in a consistent way. It should also be noted that the three-dimensional “sinuous dunes” observed by Ouriemi et al. (2009, 2010) were not observed in the present study: all bed- forms were two-dimensional (by visual inspection). Data from a total of 58 runs at several nominal concentrations (0.1% < φw < 3%) are presented and a summary table of all the runs is given in the Supplementary Material.The regions in Fig. 14 corresponding to no particle motion, rip- ples/dunes and upper plane beds are well delineated. The three time-dependent bedform types are less well delineated, but the tentative observation can be made that regular ripples are clus- tered at higher values of Ga(H/d)2 and increasing and unstable ripple-to-dune formations at lower values. This (a) is an inver- sion relative to the observations of Ouriemi et al. (2009), who ob- served “small dunes” at lower values of Ga(H/d)2 and larger, “vor- tex dunes” at higher values; and (b) highlights that they generally occur in conditions where the flow thickness becomes an impor-Upper plane bedRegular ripplesRipples to dunes, increasing period Ripples to dunes, cyclicNo particle motion
100000Repipe100001000500000 5000000Ga(H/d)2Fig. 14. Phase diagram of bedforms in pipe flow according to bulk Reynolds number, Repipe against Ga(H/d)2 . Unfilled triangles: upper plane bed; pluses: regular ripples; crosses: unstable ripple-to-dune cycles; stars: ripples that transition with increasing period to dunes; filled triangles: no particle motion. Dashed lines indicate transitions between no particle motion, time-dependent bedform fields and upper plane beds, and are fitted visually. Solid line is prediction for incipient particle motion threshold from Ouriemi et al. (2009).tant factor, although clear delineations between the two bedform types are not immediately evident. Additionally, this behavior sug- gests they are transitional between ripples and upper plane beds. There is a clear dependence of bedform type on Reynolds number, while it appears that both ripples and dunes occur in a relatively small region around Repipe = 10,000 (and thus all within a turbu- lent flow field).To reiterate, inspection of the bulk Reynolds number, Repipe , in Fig. 14 demonstrates that the flow was turbulent in all runs except for two, in both of which no particle motion was observed. In par-ticular, these two runs had Reflow = 1780 and 2680; the flow was therefore laminar and transitional/turbulent, respectively, assuming the transition in pipe flow occurs in the region of Reflow ≈ 2300– 2500 (Edelin et al., 2015; Ouriemi et al., 2009). However, the flowrates for the two laminar runs were below that for incipient parti- cle motion.A number of other observations and conclusions can be drawn from Fig. 14, which is plotted with the same variables as the phase diagram of Ouriemi et al. (2009). First, the threshold for particle motion (Repipe ≈ 6500) does not vary with Ga(H/d)2. Second, the threshold between unstable bedforms and upper plane beds ap-pears to increase with Repipe . Third, the same threshold vanishes at low flow Ga(H/d)2, although this observation is tentative as it is based on rather few data. Fourth, and most importantly, the ob- served thresholds do not appear to closely match those given by Ouriemi et al. (2009), although it is difficult to gauge the close- ness of the match as the results presented here fall into a small area in the upper limits of the parameter space investigated by Ouriemi et al. (2009). The threshold for incipient particle motionaccording to the expression given by Ouriemi et al. (2009) was calculated, namely Repipe = (2θsc/3βπ )Ga(H/d)2 = 0.0108 Ga(H/d)2, where β is a fitting constant found by Ouriemi et al. (2007) to be β = 1.85 and θsc = 0.094 as measured here, but the expressiondoes not correctly predict the observed threshold for incipient par- ticle motion, as is clear from Fig. 14. It is also noted that the instability threshold predicted by Ouriemi et al. (2009), namelyRepipe = 140φm/3βπθsc = 43.9, where φm is the maximum pack- ing fraction in the bed, with φm = 0.514 (Rice et al. 2015a), under-predicts the Reynolds number at which ripples are first observed in the present study by several orders of magnitude – and, in fact, falls below the threshold for incipient particle motion – and is therefore not included in Fig. 14.These differences are not surprising because: (a) the Ouriemi et al. (2009) model assumes viscous flow and their experiments were performed under different conditions, whereas the majority of runs were turbulent in the present study; (b) a particle flux was maintained throughout each run in the present study, whereas it was not in the Ouriemi et al. (2009) study; (c) all the bedforms observed in this study were two-dimensional, unlike the three-dimensional sinuous dunes of Ouriemi et al. (2009); and(d) Ouriemi et al. (2009) found lower plane beds – i.e. flat beds at lower Reynolds numbers than for ripples – whereas none were observed in this study; moreover, upper plane beds were observed in this study, but not by Ouriemi et al. (2009). So, the fluid and particle dynamics are clearly different, and the parameters most likely to account for the differences between this study and that of Ouriemi et al. (2009) are particle shape, size, density and particle size distribution. While Ouriemi et al. (2009) presented some data from large plastic particles of a similar size to the current study, most particle types investigated were much smaller. Additionally, all particles were spherical in shape, and had much narrower particle size distributions than those used here. These parameters will affect bed roughness and saltation-induced roughness, and in particular for broader distributions such as those used here, where bed armoring (see later discussion) by the larger particles can occur, d50 may not be the optimal measure of particle size, and d90 or another parameter incorporating the size distribution range may be more suitable (Kleinhans et al., 2017; Peakall et al., 1996). Clearly, the position of the thresholds between the various bedform types are either very different, inverted or non-existent, depending on the particle and flow parameters. The last obser- vation is that the variables used, as chosen by Ouriemi et al. (2009) – Repipe and Ga(H/d)2 – do not appear to able to capturethe universal behavior of bedforms in closed pipes, and an effortwas made to improve the parameterization of the phase diagram.
Fig. 15. Illustration of a suggested universal scaling for closed-conduit flows of any cross-sectional shape.
When comparing data from natural and laboratory systems, and from conduits with rectangular, circular and other cross-sectional shapes, it is clear from the results presented that many time- and length scales can be used, but that not all are universal because the geometry – and therefore flow field and particle concentration profiles – differ. For example, a non-zero bed depth in pipe flow modifies the chord length at the bed surface and the shape of the flow area, whereas in rectangular channel flow it does not. This is- sue is discussed in more detail, with the aim of suggesting scalings that are more universal and allow more direct comparison of data from different flow geometries at low flow rates.The most important point to note is that total bed depth, h, and pipe diameter, D, influence the flow behavior in so far as they modulate the flow area and bedform chord length. However, the ratio H/d does not appear to have a strong effect on the threshold for incipient motion, as is clear from Fig. 14. The flow does not interact with the lower, stationary part of the bed, as can be seen in the two frames in Fig. 13, and particle motion is confined to the body of the ripples.Upon first consideration, then, the three quantities that will in- fluence the behavior of the bed, and which ought to be chosen to allow comparison between various flow geometries, are (a) the fluid depth above the bed, (b) the fluid velocity at or near the bed surface, and (c) the size of the particles. However, it should be noted that in the special case of very large particles or very small flow cross-sections, the fluid depth or pipe diameter in the case of a very thin bed would be an important parameter. This is not thecase in this study, in which d50 /D ≈ 90.So, some combination of (a) H, (b) Uave, and (c) d50 may appear optimal. However, it is suggested that more physically meaningful choices can be made. In the case of (a) an equivalent fluid depth, Heq, is proposed and is as illustrated in Fig. 15. Heq is calculated by conserving the chord length, c – i.e. the cross-sectional width of the bed at its surface – and the flow area, Aflow , between flowgeometries, since these two quantities are posited as being of prin-cipal importance in terms of their influence on bedform behavior. That is,Qrole. The first is a process by which larger particles constitute the top layer of the bed, and smaller particles are thereby “armored” from the influence of fluid flow; in the second, larger particles skip over a bed of smaller particles. Indeed, in previous work on jet erosion of various mineral sediments by several of the present au- thors, it was found that d90 was a far better predictor of behavior than d50 for these reasons (Hunter et al., 2013). It is also noted that d90 , rather than d50 , was chosen by van Rijn (1984a) as a rep- resentative particle size at the bed surface when calculating C’, the particle Chézy coefficient. The resulting expressions for the trans- port stage parameter, T, and hb /H (Eq. (6)) were found to predict equilibrium bedform dimensions more accurately than others ear- lier in this section, although only reasonably accurately.
So, it is suggested that choosing Heq, Uflow and d90 – rather than, say, H or D, Uave and d50 without further consideration – has the advantages of (a) capturing all the relevant scales, and (b) al- lowing more direct comparison between data obtained in conduitsof different cross-sectional shapes. For example, in conduits with rectangular cross-sections, Heq ≡ H. When applied to a specific case, in particular the evolution of bedform height, hb , with time, t, as described in this section, the corresponding choice of param-eters would be hb /Heq against Uflow t/Heq (rather than hb /H against t as shown in Figs. 9–11). Both parameters have universal, unam- biguous meanings in many flow geometries and accurately repre- sent the physical situation in the flow, because as many important flow parameters as possible are taken in to account, and because results can readily be compared to the expressions for hb given.With the preceding arguments on universal scalings in mind, and with reference to a similar combination used by Ouriemi et al. (2009), a second phase diagram – shown in Fig. 16– was con- structed based on the combination Gaeq(Heq/d90 )2 and Reeq, which are defined as follows:eq νRe = UflowHeq , (20)90—d3 (s 1)gGaeq = ν2 . (21)Two important observations can be made from Fig. 16. Most significantly, the small, regular ripple formations seem to be clus- tered at higher values of the abscissa (i.e. larger relative fluid depths), whereas the other two variable bedform types are clus- tered at smaller values. It can also be seen from Fig. 16 that the initial relative fluid depth strongly influences the initial bedform type. For example, at a flow Reynolds number of Reeq = 104, it is conceivable that any of the five bedform types could be obtained, depending on the value of the abscissa. This dependence of bed- form type on initial fluid depth is entirely absent from the liter- ature, but has important engineering implications in terms of its possible tendency to cause blockages, plugging and flow variability generally.= = UcHeq Aflowflow, (19)So, h and D clearly have a large effect on initial bedform evo- lution since they determine the shape of the flow cross-section.where c and Aflow are calculated from the measured value of h ge- ometrically. The choice of Heq also naturally yields the second pa- rameter, (b) Uflow , as the most appropriate.The third choice to be made is a representative particle size atthe bed surface. Although d50 is the most obvious choice, it may be a poor one: if the particle size distribution is wide, then the parti- cles deposited at the surface of the bed may be significantly larger than d50, since larger particles will tend to deposit more readily than smaller ones. For this reason, d90, may be a better choice. Also, the size of particles at the bed surface may depend in a more complex way on the flow rate and ambient particle concentration, and the processes of armoring and overpassing, well known by sedimentologists (García, 2008; Raudkivi, 1976), may also play aHowever, the equivalent fluid depth, Heq, and the generalized flow Reynolds number, Reeq (which takes account of the modulated cross-sectional shape), appear to be better predictors of bedform behavior, as shown in Fig. 16, for a given initial bed depth, and once the bedform has begun to evolve. In Fig. 16, the abscissa and or- dinate incorporate Heq and Reeq and the boundaries between the various bedform types depend on both these parameters.It should be made clear that a major reason for presenting the first phase diagram was to demonstrate that (a) it is unable to rep- resent the range of results in the literature in a universal way, (b) a rethinking of the length scales – e.g. d90 in place of d50 , Heq in place of H – yields a phase diagram that has more predictive power in the sense that there is a dependence on both control
Fig. 16. Phase diagram of bedforms in pipe flow according to equivalent Reynolds number, Reeq against Gaeq (Heq /d90 )2 . Unfilled triangles: upper plane bed; pluses: regular ripples; crosses: unstable ripple-to-dune cycles; stars: ripples that transition with increasing period to dunes; filled triangles: no particle motion. Dashed lines indicate transitions between no particle motion, time-dependent bedform fields and upper plane beds, and are visually fitted.
Table 5
Analogs used to compute particle characteristics.,4Ap/π , where Ap is projected area of particle∗ As measured by the Retsch Camsizer XT.parameters, and (c) this rethinking allows for a direct comparison with a very large, mature body of earth sciences data which, al- though recorded in a range of open- and closed-conduit geome- tries and not just closed-pipe flow, are a manifestation of the same near-bed physics.Variable
Analog∗
Definition
ds
Area-equivalent diameter, xarea
Fs P
Width-to-length ratio, Fw
Krumbein roundness, Pk
c/a
Conclusions
Measurements of time-dependent bedforms produced by the deposition of solid particles from two-phase liquid-solid flows were studied using an ultrasonic echo method in a horizontal test section of closed pipe flow loop. Results were presented for settled bed thicknesses over a range of flow rates, with hysteresis behavior in plane beds and ripples also considered. The evolution and scal- ing of bedform heights were then investigated. In the concluding part of the results, data gathered in a wide range of experiments were used to derive phase diagrams of bedforms in closed pipes in terms of several dimensionless numbers in which the thresh- olds between incipient particle motion and different bedform types were established.
Most importantly, both ripple- and dune-type bedform fields were observed under certain flow conditions, sometimes as two distinct temporally varying modes within single runs. This behav- ior has not been reported before, and it is thought that it may be due both to the particular particle size used in this study and to the irregular shape and roughness of the particles. Correlations for equilibrium bedform dimensions, such as those devised by García (2008), Julien and Klaassen (1995) and van Rijn (1984a), were of limited value, although the last was found to most accurately pre- dict the height of the evolved bedforms in this study.
Several non-equilibrium phase diagrams were generated cate- gorizing formations as stable ripples, transitioning ripple-to-dune
bedform fields and cyclical, alternating ripple-and-dune forma- tions, as well as stable upper-plane beds. It should be reiterated that this study was not intended to be of equilibrium or saturation bedform dimensions – the distinction being that the former is ob- tained under clear-fluid conditions whereas the latter is obtained with a constant particle flux (Edelin et al., 2015) – but rather of the initial behavior beginning with flat, planar beds. This choice was driven by industrial concerns, specifically the potential prob- lems experienced upon start-up of machinery, for example, and in the resuspension and transport of settled solids. Phase diagrams would certainly be expected to be significantly different in equilib- rium and saturated conditions, as they are known to be in natural and flume/open-channel flows. However, this study addresses only the case of initial behavior. In addition, it is noted that bedforms under equilibrium and saturation conditions in closed pipes have received very little attention, and the authors are aware of only one such study (Edelin et al., 2015).
No single model, either from the sedimentology or the engi- neering/fluid mechanics literature, was able to fully account for the observed bedform types, in terms of a phase diagram or other- wise. It was suggested that both a full set of initial conditions –
at least d90 , hb, Heq, t, Uflow and ν, where He is an equivalent fluid
depth, as illustrated in Fig. 16– as well as the magnitude and type of perturbation applied to the bed, be included in any dimensional analysis that is performed in order to derive a universal scaling for bedform dimensions. Any model must necessarily incorporate the dynamic nature of bedforms, including the hysteretic, path- dependent behavior described here, and the influence of initial and resultant bed and fluid depth and changes in flow rate, which are strongly linked. Further data would, however, be necessary in or- der to undertake such a task. The effect of the shape of the pipe cross-section must also be taken into account in future studies, and
suitable scalings (e.g. Uflow in place of Uave; H or He in place of D; see Section 4.5) should be chosen and justified over a range of flow geometries and particle types.
Acknowledgements
The present study is based on part of the Ph.D. thesis of H.
P. Rice (“Transport and deposition behavior of model slurries in
(Syvitski, 2007) and has traditionally been assessed visually to a precision of ± 0.5, and
c
Fs = √ab , (28)
where a, b and c are the longest, intermediate and shortest axes of the particle. The expression for w is as follows:
w3
closed pipe flow”, University of Leeds, 2013). The authors wish to thank the Engineering and Physical Sciences Research Coun-
w∗ = R310R1 +R2 =
gν(s − 1)
, (29)
cil for their financial support of the work reported in this paper under EPSRC Grant EP/F055412/1, “DIAMOND: Decommissioning, Immobilisation and Management of Nuclear Wastes for Disposal”. The authors also thank Peter Dawson, Gareth Keevil, Russell Dixon,
where w∗ is the settling velocity and R1 , R2 and R3 are as fol-
lows:
3
4
R1 = −3.76715 + 1.92944(log dw ) − 0.09815(log dw )2
Rob Thomas and Helena Brown for their technical assistance, and Olivier Mariette at Met-Flow, Switzerland, for his support and ad-
vice.
−0.00575(log dw ) + 0.00056(log dw ) , (30)
R = hlog³1 − 1 − Fs ´− (1 − F )2.3tanh(log d − 4.6)i
Appendix
Expressions for bed shear stress and Shields number
2 0.85 s w
2
+0.3(0.5 − Fs )(1 − Fs ) (log dw − 4.6), (31)
h ³ Fs
´i{1+(3.5−P)/2.5}
The bed shear stress, τ b, can be written as follows:
τb =
f (Re∗ ), (22)
ρf ³ Q ´2
2
A
flow
R3 =
0.65 − 2.83 tanh(log dw − 4.6)
. (32)
Here dw is a dimensionless particle diameter based on ds, the di-
ameter of a sphere of equivalent volume, such that:
where f is the Darcy friction factor as a function of Re∗, the
Reynolds number based on a length scale, D∗, computed numer- ically by Peysson et al. (2009) and referred to as the equivalent
dw = d3
g(s − 1)
ν2
. (33)
diameter such that:
·³ 8 ´12
¸1/12
In practice, the Retsch Camsizer XT was used to measure ds, Fs
s
and P indirectly, using the analogs described in Table 5 below, and
f = 2
Re∗ D∗Q
+ (A + B)−1.5
, (23)
it was assumed that (a) ds = xarea, (b) Fs = Fw and (c) P = 6Pk , since
0 < Pk < 1 and 0 < P < 6. This method is presented as a powerful tool for quantitative assessment of the effect of particle sphericity
Re∗ = νA
flow
, (24)
and roundness on settling and resuspension.
½ ·³ 7 ´0.9
ε ¸¾16
Supplementary materials
A = −2.457 ln
B =,
³ 37530 ´16
Re∗
Re∗
+ 0.27 D∗
, (25)
, (26)
where ε is the roughness length. D∗ is the diameter of an equiv- alent pipe having the same shear stress at the wall as at the bed surface in a flow with a bed depth of h/D; values of D∗ as a func- tion of h/D are given by Peysson et al. (2009). The Shields number can then be calculated from the bed shear stress, τ b, and other known flow variables via Eq. (12).
Alternatively, the critical Shields number, θ sc, i.e. that at the threshold of particle motion, can be estimated using a commonly cited expression, as follows (Edelin et al., 2015; Soulsby, 1997):
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