CC BY
46
университета в о д н ы х коммуникаций
13. Lisle T. E. (1997) Evolution of a sediment wave in an experimental channel / T. E. Lisle, J. E. Piz-zuto, H. Ikeda, F. Iseya and Y. Kodama // Water Resour. Res. — 1997. — № 33 (8) — P. 1971-1981.
14. Maddux T. B., Nelson J. M. and McLean S. R. (2003) Turbulent flow over three-dimensional dunes: 1. Free surface and flow response // J. Geophys. Res. — 2003. — № 108. (F1, 6009, doi: 10.1029/2003JF000017).
15. Nakayama S., and Shimizu Y. (2001) Numerical calculation of suspended sediment over bed-forms // Proc. Int. Conf. Riv. Coast. Morph. Dyn., Obihiro. — Japan, 2001. — P. 217-224.
16. Nelson J. M. (2005) Computing flow and sediment transport over bedforms / J. M. Nelson, A. R. Burman, Y. Shimizu, S. R. Mclean, R. L. Shreve and M. Schmeeckle // Proc. Int. Conf. Riv. Coast. Morph. Dyn. (submitted). — 2005.
17. Nelson J., and J. D. Smith (1989) Mechanics of flow over ripples and dunes // J. Geophys. Res. — 1989. — № 94 (C6). — P. 8146-8162.
18. Onda S. and T. Hosoda (2004) Numerical simulation on development process of dunes and flow resistance // Proc. Int. Conf. Fluv. Hydraul. — Napoli, Italy, 2004. — P. 245-252.
19. Paarlberg A. J. (2006) Modelling morphodynamic evolution of river dunes / A. J. Paarlberg, C. M. Dohmen-Janssen, S. J. M. H. Hulscher, J. Van den Berg and A. P. P. Themes // Proc. of River-flow. — 2006. — P. 969-978.
20. Schindler R. J. and A. Robert (2005) Flow and turbulence structure over ripple-dune transition: an experiment under mobile bed condition // Sedimentology. — 2995. — № 52. — P. 627-649.
21. Shimizu Y. (2002) A method for simultaneous computation of bed and bank deformation of a river // Proc. Int. Conf. Fluv. Hydraul. — Louvain, Belgium, 2002. — P. 793-801.
22. Shimizu Y., Hirano N. and Watanabe Y. (1996) Numerical calculation of bank erosion and free meandering, Ann // J. Hydraul. Eng. — JSCE, 1996. — № 40. — P. 921-926 (in Japanese).
23. Shimizu Y., M. W. Schmeeckle and J. M. Nelson (2001) Direct numerical simulation of turbulence over two-dimensional dunes using CIP methods // J. Hydrosci. Hydraul. Eng. — 2001. — № 19 (2) — P. 85-92.
24. Sugiyama H., Akiyama M. and Matsubara T. (1995) Numerical simulation of compound open channel flow on turbulence with Reynolds stress model // J. Coast. Envir. Eng. — 1995. — № 515/II (31) — P. 55-65 (in Japanese).
25. Van Rijn L. C. (1984) Sediment pick-up funtion // J. Hydraul. Eng. — ASCE, 1984. — № 110 (10). — Р. 1494-1502.
TWO-PHASE BOUNDARY LAYER MODEL TESTED AGAINST COUETTE FLOW
MEASUREMENTS
ДВУХФАЗНАЯ МОДЕЛЬ ПОГРАНИЧНОГО СЛОЯ, ПРОТЕСТИРОВАННАЯ ^69
С ПОМОЩЬЮ ПОТОКА КУЭТТА
Arno Talmon,
Delft University of Technology
о X 2
In the hydraulic pipeline transport of concentrated solid-liquid mixtures there is a need for improved physics-based friction laws. For flow velocities higher than the deposit-limit-velocity the equivalent liquid hypothesis, or variations on it, are often applied. For these conditions, and for the suspension flow above deposits, a boundary
II университета
'ЖУРНАЛ водн ы х / / коммуникации
layer theory for highly concentrated solids-suspensions including grain shear stresses is proposed. Particles influence the velocity profile close to the wall (laminar sub-layer and buffer-layer) and influence wall shear stresses. In this paper the boundary layer theory is tested against controlled Couette flow measurement (Bagnold's measurements). Calculations indicate that shear rates near the wall were higher than the imposed gap-averaged shear rate, and that shear rates were lower in the bulk of the flow. As a consequence the numerical value of the constant in Bagnold's grain stress formula had to be increased by a factor 2.
При трубопроводном транспорте концентрированных смесей жидкости и твердых веществ существует необходимость уточнения физических законов трения. При скорости потока выше, чем предельная скорость образования наносов, часто применяют гипотезу эквивалентной жидкости. В этих условиях при наличии взвесей в потоке предлагается теория пограничного слоя для высоко-концентрированных твердых взвесей с учетом напряжений сдвига. Мелкие частицы влияют на эпюру скорости близко к стенкам (ламинарный подслой и буферный слой), а также на напряжение в стенках. Теория пограничного слоя проверена с помощью замера контролируемого потока Куэтта (замеры Багнольда). Расчеты показывают, что скорость сдвига около стенки выше, чем усредненная скорость сдвига, и что эти скорости ниже в основном массиве потока. Как следствие, цифровое значение константы в формуле напряжения Багнольда необходимо увеличить на 2.
Key words: boundary layer flow, hydrotransport, grain stresses, solid-liquid mixtures.
Ключевые слова: поток пограничного слоя, гидротранспорт, напряжения грануляции, смеси жидкости и твердых веществ.
DHIS paper was presented at the 14th International conference on "Transport and Sedimentation of Solid Particles". The conference was held in the Saint Petersburg State Mining Institute from 23 till 27 June 2008. The conference is a forum in the field of pipeline transport, of mainly solid, friable or liquid products, and is held in various countries of the world biannually. The conference is devoted to modern hydraulic transport, a progressive economical mode of transport, characterized by: universality, absence of cargo loss in the process of transportation with full mechanization and automation of labour consuming cargo- handling operations and others.
The 1st international conference was held in 1971. This conference took place in Wroclaw Agricultural University (Poland) by the initiative of a group of leading scientists from Poland and Czechoslovakia, with the aim to promote wide exchange of scientific information and familiarization with developments in that field. Later on
psj
g the conference was held in the Czech Republic,
| Poland, Belgium, Germany, Georgia. In 2008, the
E°° ^ conference was held in Russia for the first time.
Representatives from leading countries in hydraulic transport took part in the conference: Australia, Belgium, Bulgaria, Czech Republic, France, Great Britain, Germany, Georgia, Israel, Italy, the Netherlands, Poland, Ukraine, Russia, Sweden and South Africa.
60 scientific reports were listened to at 10 sessions during the conference week. They were on current problems of transportation of two-phase and multiphase flows moving solid, friable liquid composition and mineral resources: coal, sand, gravel, salt solutions, construction mixtures and others: concentrating mills waste, ash and slag from thermal power plants in dumps, municipal drainage and waste; barren rock to the place of storage at the mining enterprises, including issues on dredging and issues of pipeline transport for oil, oil products and natural gas.
The author acknowledges the organizers of the conference for their kind permission to reproduce the paper that the author presented at the conference. For further reading, the reference to the conference proceedings is: J. So-bota, S. Ivanov and V. Alexandrov, 2008, Proc. 14th Int. Conf. on Transport & Sedimentation of Solid Particles, Saint Petersburg State Mining Institute (Technical University), ISBN 978-594211-350-6.
1. Introduction
Present friction models for solid-liquid mixtures are based on empiricism, but scale effects are likely to play a role. It is recognized that boundary layer flow determines friction, and that boundary layer flow is independent of the scale of the pipeline (though Reynolds number effects might be present). Concentration measurements
Fig. 1. Principal configuration of concentration profile, velocity profile and shear stress distribution in Couette
flow of highly concentrated solids-suspensions
have revealed that de-mixing occurs at the bottom of open channel flows: Shook et. al. 1968. Micha-lik 1973 measured diminishing concentrations near the invert of hyper-concentrated pipe line flows. In case of high flow velocities. friction factors have been found that are smaller than in clear water flow (compared on basis of equivalent fluid hypothesis, see Carstens & Addie 1981 and Wilson et.al. 2005). Wilson & Sellgren 2003 have attributed this to an off-the-wall lift force being effective in the 100-600 micron region, lowering the concentration at the wall that leads to a decreased friction factor. In their analysis this is brought about by a retardation of particles by curvature of velocity profile in combination with a Kutta-Zhukovski circulation lift-force.
The present contribution is centred on a mechanism of dispersive grain stresses that. in combination with a viscous lift force produces an off-the-wall force. These mechanisms are quantified and lead to non-uniform mixture concentrations near the wall and an altered shape of the velocity profile close to the wall (which affects the friction coefficient). In this paper the model is tested against controlled Couette flow measurement (measurements of Bagnold 1954). The expected flow pattern of a highly concentrated solids-suspension in Couette flow is sketched in Figure 1.
2. Boundary layer Model incorporating grain shear stresses
In order to quantify hydraulic friction for various concentrations and particle diameters a mixing length based boundary layer model has been devised. Modelled fundamental processes are: Bagnold grain stresses (normal and shear
yHMBepcMTeTa b o a H bl x KOMMyHMKauMM
stresses in grain-inertia regime), hydrodynamic lift (viscous lift force, Saffman 1965), an adapted low-Reynolds mixing length model (von Driest theory; for velocity profile, see Schlichting 1960), Boussinesq gradient diffusion hypothesis for concentration profile (no influence of stratification), and hindered-settling of particles.
The fundamental equations are solved it-eratively: a 2-D force balance for particles, a 1-D convection-diffusion equation for the calculation of the concentration profile in the boundary layer, and a 1-D mixing length model for velocity profile, including the influence of grain stresses. A definition sketch of the flow close to the wall is given in Figure 2.
Fig. 2. Sketch of boundary layer profiles with grain shear stresses
2.1. Velocity profile modelling
Shear stresses by clear fluid turbulence and shear stresses by inertial grain stresses are respectively:
\2
2
mm
P ml
du ~dz
and
(1)
T ' = № Pm &d)2
with:
du dz
V
d = particle diameter; L = mixing length; u = flow velocity; z = cross stream coordinate; ^ = dynamic granular friction coefficient: !=x '/o ';
t ' = grain shear stress; o ' = normal grain stresses; rt = turbulence shear stress; a. = constant in Bagnolds formula for normal stresses in grain inertial regime; pm = mixture density; X = linear solids concentration (grain diameter/shortest distance between grains):
B0
B-
X 2
T
yHHBepcHTeTa B o a H bl x KOMMyHMKauMM
CM £ u
X
1
(Cm / C)1/3 - 1
(2)
with: c = volumetric solids concentration;
c = maximum volumetric solids concen-
m
tration.
The shear stress in the wall region of the boundary layer is assumed constant, in analogy to clear fluid boundary layer modelling. The shear stress is modelled as a superposition of viscous, turbulent and inertial grain stresses:
— 2 du 2 du 2 T =p mU* mL(— )
dz dz
with: L = lm + |ia, (Xd )2,
(3)
with: L = length scale in momentum exchange process;
n = dynamic mixture viscosity; t = total shear stress; pm = mixture density at the edge of the boundary layer;
u* = ^x/pm = friction velocity.
The dynamic viscosity of the mixture is modelled as a function of local solids concentration (Leighton & Acrivos 1986):
n.
n w
1 + -
1.5c
2
1 - c/0.6
(4)
with: nw = liquid dynamic viscosity (water).
The mixing length (lJ is modelled analogous to the low-Reynolds theory by von Driest, see Schlichting 1960. For flows bounded by two facing walls, such as the Couette flow of Section 3, the low-Re mixing length is modelled by:
lm — K Z (1 -- )(1 - c h
zu* 1 ( h-z )u* 1
- c v A ), (5)
with: h = depth of flow;
k = 0.4 = von Karman constant; A = 26 (von Driest) constant. For application to particle laden flows, the kinematic mixture viscosity v at the edge of laminar sublayer (z+=11) is substituted in the exponential term of Eq.(5). Stratification influences on the mixing length have been neglected.
The principal properties of the model are: 1. In the outer region of the boundary layer flow, the velocity gradient is dominated by tra-
ditional mixing length (grain effects negligible). Therefore, remote from the wall, the velocity gradient is the same as for clear water flow (at the same friction velocity u*).
2. At the wall (laminar sub-layer and part of the buffer layer) where depleted regions may occur, the density and mixture viscosity decrease. Since there is a density difference between the outer edge of the boundary layer and the fluid near the wall, velocity gradients near the wall are higher for the same u* (see viscous term of Eq. 3). Consequently the friction factor f=8 (u*/U)2 decreases when grain stresses are negligible.
3. Grain stresses on the wall, on the other hand, will lead to an increase in friction factor.
2.2. MecHanical balance of particles
There are four forces acting on the particles: particle-weight, buoyancy, lift and grain-stress gradients. In shear flow, with a co-rotating slipping particle, the viscous lift force acting on a single particle is (Saffman 1965):
F-kear =a d 2 P Vif dZ
(6)
with: Vt= slip velocity particle to fluid;
a = constant (~20.5 according to Saffman
1965).
Saffmann's equation applies to a single particle in clear fluid. For our concentrated mixture conditions the mixture viscosity is substituted. Grain shear stresses with the wall will slow down the particles a little bit. Consequently particles will slip relative to fluid (V), and the lift force appears. In flow direction, the movement of a particle is driven by the acceleration:
1 - c
dT'
ax =
c(P s -P w ) dz
(7)
with: ps = solids density;
pw = liquid density (water). Perpendicular to a wall (located beneath the flow), the movement of a particle is driven by the acceleration:
a, = (1 - c)( g +---—--Flshear ),
c(Ps -Pw) & (ps -pw)V
n
(8)
with: V = — d3, 6
with: ax and az = accelerations acting on grains;
g = gravitational acceleration;
V = volume of one particle. Body forces (own weight) and external forces (grain stresses and lift) are causing particle movement. In equilibrium these forces are transferred, via fluid drag, to the surrounding fluid. As a consequence a conglomerate of particles being subject to these forces is changing the pressure gradient of the surrounding fluid. Since for a single particle this pressure gradient acts analogous to a buoyancy force, the external forces in the momentum balance of a single particle have been corrected by a factor (1-c) in Eq.(7) and Eq.(8). In the calculation of the relative velocity of the particles to the flow, the driving force is V(ps- pw)a. After the relative velocity of a single particle is calculated, this velocity is corrected with the hindered settling formula. The hindered settling exponent is lowered on input because one (1-c) term is already accounted for in the equations above.
2.3. Calculation of concentration profile Momentum and mass are transferred by turbulence and inertial particle movements. The solids concentration profile is calculated numerically on basis of exchange coefficients and relative velocity of the particles. The mass-exchange coefficient is modelled on basis of mixing length theory, in this case consisting of a superposition of clear fluid turbulence and grain inertial effects:
K = PtL2 ^ [m2/s], (9)
dz
with: Pt=1.4 (reciprocal of turbulent Schmidt-Prandtl number: 1.4 in neutral conditions).
For distance closer than one particle diameter from the wall, the formulation of Acrivos & Chang 1986 for the transition from continuum systems to discrete systems is used:
2 z o 2 z
c_ )2(3 - 7»
4
(10)
with: c = solids concentration at a distance of one particle diameter from the wall.
This volumetric concentration (c) is strictly geometrical, and approaches zero at the wall. Eq. (10) is used in combination with Eq.(4) for the calculation of mixture viscosity in the laminar sublayer and buffer layer. The grain stresses on the contrary are modelled constant over a distance
yHHBepcHTeTa b o a H bl x KOMMyHMKauMM
of one grain diameter from the wall, because of the discrete physical dimensions of this stress producing mechanism. Grain stress calculation is thus not influenced by Eq.(10).
3. Boundary layer theory applied to Couette data
Couette flow is a fundamental flow condition that is suited for verification/validation of the model. Ideally, the shear stress across Couette flow is constant, just as in boundary layer theory. The original Couette flow data of Bagnold 1954 is re-evaluated. Bagnold conducted measurements on highly concentrated Couette flow in a concentric cylinder apparatus of which the outer cylinder was driven by an electric motor (D = 114 mm, gap width = 10.8 mm, diameter neutrally buoyant wax spheres = 1.32 mm). The slight variation of shear rate across the annulus in Bagnold's experiments is neglected. Bagnold measured rotational velocities, total shear stress (= fluid + grains) and normal stresses (= grain stress). The boundary layer theory of Section 2 gives these stresses on output. The viscosity of the employed fluids was 0.001 and 0.007 [Pa s], and the density was equal to water. Volumetric solids concentrations were in the range 13 < c < 62 %. Consequently the di-mensionless particle scale in the experiments was 25 < d+ < 320. This shows that the particles were in the buffer region and in the fully turbulent layer of boundary layer flow. These particles do not fit in the laminar sublayer (z+ < 11). The thickness of the Couette flow in these experiments is in the range of: 200 < h+ < 2500, indicating that there was not much logarithmic velocity profile (logarithmic profile starts at z+ > 70).
In Bagnold's original derivation of his formulae for grain stresses, it was assumed that the shear rate is constant over the depth of the flow. In fact this will only be true for mixtures with constant viscosity across the flow. According to the boundary layer theory of Section 2, however, the solids concentration will be lower in the wall region, becoming higher away from the wall. As a consequence shear rates are higher at the wall and lower in the middle of the flow, see Figure 1.
Based on overall shear rate Bagnold found:
a. = 0.04. The ratio of total shear stress and normal
i
grain stress (=dynamic friction coefficient varies gradually between the macro-viscous regime
BO
B"
X 2
университета в о д н ы х коммуникаций
(N < 40, ^ ~ 0.75) and the inertial regime (N > 450, ^ ~ 0.2), see Hunt et. al. 2002. By neglecting the fluid shear stress, the dynamic friction coefficient in our model is approximated by:
|l = 0.75
1 + 0.2(N )15 100
1 + 0.75(N )15 100
(11)
CM ж и
with: N = ,
V dz
with: N = Bagnold number.
The simulations were done on basis of
normalised mixture velocity u/u* [-]
0 2 4 6 8 10 12
N 2000
ai я с
1500
<3
о и
и 1000
й 500 с 01 Е
=5 0
г
' У /
; (
J / \
.j
2000
1500
1000
500
0
0 0.2 0.4 0.6 0.8 1
solids concentration & normalised grain shear stress [-]
Fig. 3. Calculation results for Couette flow at: c = 55 %, z+ = 2244, d+ = 270, a = 0.08
^ max '
■D
О □
и
known shear stresses measured by Bagnold, and the velocity difference between the concentric cylinders was calculated. The input parameters to each calculation were: d+, g = 0, c, c = 0.74, h+ and the hindered settling exponent. Outputs are the dimensionless velocity difference Au/u* and calculated profiles.
The total number of grid points in the calculation is 420. In both wall regions, up to z = d, the grid interval was Az+ = 3. The remaining grid points are situated equidistantly in the bulk of the flow.
Fig. 3 shows the results of the calculated profiles within the sheared gap of one of Bagnold's Couette shear flow experiments. The figure shows that grain shear stresses are modelled constant over the last row of particles along the walls. The flow velocity varies strongly over a distance of one particle diameter from the wall because mixture viscosity decreases to that of the carrier fluid at the wall (Acrivos & Chang function).
Only the coefficient of Bagnold's normal stress equation (a) is considered as a calibration parameter. An optimal result is obtained for a. = 0.08, see Figure 4, which is double the value Bagnold found when he defined the shear rate as du/dz = Au/h. In the present analysis, grain stresses have been related to local velocity gradients in the flow. According to this result, at the same shear rate and mixture conditions, the normal grain stresses are about a factor two greater than according to Bagnold's original formula.
tau'/tau solids conc. - u/u*
a =0.12
a = 0.08 a = 0.04
measured u*/u (at A< 10) [-]
Fig. 4. Calculated dimensionless friction velocity versus measured dimensionless friction velocity in Bagnold's Couette tests (data points: grain inertial regime and c < 55 % (1 < 10))
4. Conclusions
In Bagnold's experiments a depletion of the mixture occurred close to the wall, leading to higher shear rates close to the wall. In the main body of the flow the shear rate is consequently lower. Consequently, the coefficient of the Bagnold equation for the grain inertia regime had to be increased a factor 2.
For application of the boundary layer theory in pipes, the concentration profile in the mid-region of the pipe has to be calculated too. In case of sand-
yHHBepcHTeTa b o A H bi x KOMMyHMKauMM
water mixtures, near the deposit limit velocity, the flow will experience suspension-stratification influences. Its relation to the densimetric Froude number of pipe-flow was elaborated on by Talm-
Acknowledgement
on 2006.
The dredging companies Boskalis and van Oord are acknowledged for associated contract research furnished to Deltares through their joint SSB-re search-program.
References
1. Acrivos A. and Chang E. (1986) A model for estimating transport quantities in two-phase materials // Phys. Fluids. — 1986. — Vol. 29 — № 1. — P. 3-4.
2. BagnoldR. A. (1954) Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear // Proc. Royal Soc. — L., 1954. — Vol. 225. — P. 49-63.
3. Carstens M. R. and Addie G. (1981) A sand-water slurry experiment // J. Hydraulics Division ASCE. — 1981. — Vol. 107, HY4. — P. 501-507.
4. Hunt M. L., (2002) Revisiting the 1954 suspension experiments of R. A. Bagnold / M. L. Hunt, R. Zenit, C. S. Campbell and C. E. Brennen // J. Fluid Mech — 2002. — Vol. 452. — P. 1-24.
5. Leighton D. and Acrivos A. (1986) Viscous resuspension // Chemical Engineering Science. — 2002. — Vol. 41. — P. 1377-1384.
6. Michalik A. (1973) Density patterns of the inhomogeneous liquids in the industrial pipe-lines measured by means of radiometric scanning // La Houille Blanche. — 1973. — № 1. — P. 53-57.
7. Saffman P. G. (1965) The lift on a small sphere in a slow shear flow // J. Fluid Mech. — 1965. — Vol. 22. — Pt 2. — P. 385-400.
8. Schlichting H. (1960) Boundary layer theory. — McGraw-Hill; N. Y., 1960.
9. Shook C. A. (1968) Flow of suspensions of solids in pipelines / C. A. Shook, S. M. Daniel, J. A. Scott & J. P. Holgate // The Canadian Journal of Chemical Engineering — 1968. — Vol. 46, № 4. — Pt. 2: Two Mechanisms of Particle Suspension. — P. 238-244.
10. Talmon A. M. (2006) Modelling the influence of suspension stratification on turbulent mixing in pipelines // 13th conf. Transport and Sedimentation of Solid Particles. — Tiblisi, 2006. — P. 314-321.
11. Wilson K. C. (2005) Slurry Transport using Centrifugal Pumps / K. C. Wilson, G. R. Addie, A. Sellgren and R. Clift. — 3rd ed. — Springer-Verlag, 2005.
12. Wilson K. C. and Sellgren A. (2003) Interaction of particles and near-wall lift in slurry pipelines // J. Hydraulic Engineering. — 2003. — P. 73-76.
B0
B-
o X 2
vE.
