ON THE TURBULENT TRANSPORT OF
A HETEROGENEOUS SEDIMENT
By J. N. HUNT
(Department of Mathematics, University of Beading)
[Received 30 April 1968]
SUMMARY
The transport of sediment in steady turbulent flow is described by means of a
diffusion equation. It is shown how the inolusion of sediment volume in the continuity equation leads to a distribution of concentration with depth tending to unity
at the bed. With a mixture of grain sizes in suspension, the smaller particles can
increase in concentration upwards, while the heavier particles are necessarily
transported mainly near the bed, in accordance with observations.
1. Introduction
IN a recent illuminating review of the problem of particle motion in
turbulent flow, Batchelor (1) derives a number of conclusions regarding
the distribution of particle concentration in turbulent shear flow from the
solutions of the approximate sediment diffusion equation
^+wC
0,
(1)
where C(z) is the particle concentration at height z, e(z) a diffusion coefficient, and w the particle settling velocity. The usual derivation of this
equation contains the implicit assumption that its region of validity lies
within that range of low particle concentrations for which inter-particle
collisions may be ignored. However, some of the consequences of equation
(1) relate to the region near the bed of a channel where the solutions for
C(z) invariably become infinite. Physically, the volumetric concentration
should not exceed unity, of course, BO that there is a risk of some inconsistency in using equation (1) in such a region.
As was shown some time ago by Hunt (2), a careful treatment of the
continuity equation leads to a more general diffusion equation whose
solutions tend to a finite value at the bed. The central feature of that
treatment was the assumption that the volume occupied by the sediment
particles must be taken into account, particularly near the bed. It is by
no means obvious that the neglect of particle collisions is equivalent to
the neglect of particle volume. Some physically realistic results follow
[Quart. Jooro. M«ch. and Applied Math., VoL XXII, Pt. a, i»6g]
236
J. N. HUNT
from the hypothesis to be made in this paper, namely that collisions may
be ignored, while retaining the terms in the continuity equation due to
the volume of fluid displaced by the particles. Some of the questions
considered by Batchelor (1) will therefore be re-examined using the more
general diffusion equation.
First, however, it will be convenient to derive general equations to
describe the diffusion of a heterogeneous sediment, from which some
results for a homogeneous sediment may be obtained as a special case. A
result of some significance shows that in a suspension consisting of a
mixture of particle sizes, the fine portion may increase in concentration
upwards over an appreciable depth of the flow. Although this phenomenon has been frequently observed (4, 5), it was previously thought to
be inconsistent with turbulent diffusion, and to depend on particle
collisions at high concentrations. It will be shown that, on the contrary,
this behaviour is consistent with turbulent diffusion under certain
circumstances provided that the volumes displaced by each size fraction
of the sediment are considered.
Although the existing method of predicting the sediment distribution
is here extended to include a mixture of sizes, as well as sediment volume,
use is still made of the traditional method of supposing the eddy diffusivity
to be a prescribed function of height above the bed. We know the
diffusivity to be dependent on sediment concentration however, but until
accurate quantitative observations are available it seems permissible
to neglect this variation.
2. Kinematic relations
The basic assumption to be employed throughout this discussion of
sediment transport in turbulent flow states that particles are convected,
and also are diffused according to the classical gradient diffusion law.
The flux of sediment of a given size, whose concentration is cr, is therefore
given by
Pr = Ur<V-er S™& Cr,
(2)
where eT is the diffusion coefficient for this particular size fraction, and
u r is the sediment convection velocity which differs from the mean fluid
velocity only by a settling speed wr characteristic of this size fraction. It
is not necessary to assume that wT is the settling speed in still water.
The corresponding flux vector for the flow of water is
where the summation is overall the sediment size fractions, and where
TURBULENT TRANSPORT OF HETEROGENEOUS SEDIMENT 237
ea is the diffusion coefficient appropriate to the water and V the mean
velocity of the water.
Under general unsteady conditions, the continuity conditions are
—!• + div p r = 0,
dt
and
r=
1,2,3,...
div(q+2 pr) = 0.
(4)
(5)
For steady conditions however, these reduce to
divpr = 0,
and
r = 1, 2, 3,...
div q = 0.
(6)
(V)
The remaining relation between the p r and q is given by
ur = v - w x
(8)
where z is the unit vector in the upward vertical direction.
Substituting from (2) and (3) in (6) and (7), and making use of (8), we
have the continuity equations
div{(v—w,z)cr—er grad cT) = 0,
(9)
div{v(l-2 cr)+€w grad J cr} = 0,
(10)
for r = 1, 2, 3,... When the distribution of each size fraction is a function
only of the vertical coordinate z, and there is no net flux of sediment in the
vertical direction, then (9) and (10) reduce immediately to
(vt-wr)er-er%I
= 0,
r = 1,2,3,...
(11)
Oz
and
equations given in the reference cited (2). In this simple one-dimensional
case, (11) and (12) follow immediately from equations (2) and (3). Now
a simple re-arrangement of the system (11) and (12) leads to the equations
c
r-a
oz
iTi-r
,
T-
.
^^
r ^ ; — v>
r — 1, 2, 3 , . . .
(13)
and
v
,=
The significance of equation (13) is more clearly apparent when we make
238
J. N. HUNT
the simplifying assumption that all the er are equal to e, say. Then
1W
JJ%
% ) )= 0
^
(15)
and
Further simplification occurs when
namely
'l = 0)
In particular, for a homogeneous sediment with a concentration c(z),
equation (13) becomes
v-e,)}^+(l-c)aw
= 0
(19)
dz
(2, equation (28)). Only when two additional assumptions are made does
this equation reduce to (1), the simple diffusion equation first given by
Rouse (3), namely
.
h
J
(a) eu = e, or c( e .- 6 f ) <e.)
and
(20)
(b) c < 1.
J
3. Characteristics of a heterogeneous suspension
Customarily, the simple diffusion equation for a homogeneous sediment,
equation (1), leads to the conclusion that
- < 0 for all z
(21)
dz
since e > 0, and also c —>• 0 at a free surface where e = 0. In the case of
a mixed sediment, equation (1) has been assumed to hold for each size
fraction independently, provided that each concentration is low enough.
If this were true, it would then follow that for all z,
|r<0,
dz
r=l,2,3,..
That this is not necessarily correct, follows immediately from equation
(13), or with the simplifying assumptions referred to, from equations (15)
and (17). For the concentration of a particular size fraction c, to increase
TURBULENT TRANSPORT OF HETEROGENEOUS SEDIMENT 239
with height,
(jGi
OZ
and
„
W <
< €+u -IfVc
or
(22)
„
i < J, w<cr ( w h e n c » = «,)•
(23)
From the inequalities (22) or (23), it can be seen that a low concentration
of a heavy size fraction can interact, through the continuity equation,
with a very fine size fraction with a small settling velocity, to cause its
concentration to increase with height. This behaviour moreover is not
dependent on the precise behaviour of et or ew as a function of height.
A physical interpretation of the inequalities (22) and (23) may be
derived from a comparison of equations (1) and (17). In equation (1) the
downward settling wC is equated solely to the upward diffusion — e dC/dz.
In equation (17) however, with r = i, downward settling wfii together with
downward diffusion e dctjdz is balanced by an upward flux caused by the
volume displaced through settling of all size fractions, that is ct J cTwr.
From (22) and (23) it can be seen that only the heavier fractions wT ^> w{
can bring about this effect on c{ that is dc{jdz > 0, and then only in
regions where 2 wtcr ^ sufficiently large. This 'squeezing' process is
somewhat analogous to phenomena in non-equilibrium thermodynamics
in which concentration gradients may be brought about by a pressure
gradient, or by a temperature gradient (Soret effect).
Various experiments have shown eJeK = /? to be approximately a
constant; Ismail (6) obtained 1-3 < /? < 1-5 and Singamsetti (7)
1-2 < /5 < 1-45. The inequality (22) then becomes
w
As a numerical example, we may consider the somewhat artificial case
of a suspension of two sizes of sand consisting of grains of 1 mm diameter
{wl B» 10 cm/s) and grains 10 ^m in diameter (w2 <%* 10~8 cm/s). From
inequality (23), it follows that the concentration of the finer material
increases with height wherever the concentration of coarse material
exceeds 0-1 % by volume, when /? = 1, or 0-14% when /? = 1-4.
This condition can be satisfied over an appreciable portion of the depth
of flow, but cannot hold very close to a free surface where c = 0, and all
cr = 0. This condition at a free surface may be deduced from equation
(17) for example, for each size fraction, or alternatively from a summation
240
J. N. HUNT
of (17) over all r:
e—
When 2 <V < 1, and its gradient remains finite, each term in the series
2 wTcr must vanish where e = 0, and so all cT = 0. The second solution
when e = 0 i s ] £ c r = l which refers to the lower boundary, that is the
bed z = 0.
Also from equation (24) we see that d 2 crl^z < 0 f° r &U z> 8° that the
total concentration always decreases upwards, even though the concentrations of individual size fractions do not necessarily do so.
It is of interest to note that the non-linear system of simultaneous
first-order equations (17) for the functions cr(z) possess an explicit solution
in terms of a 'universal function' y(z) where
fe (25)
J e(z)
e(z)
namely
c
tz) =
1
cf(a)exp{tgr(y(tt)-y(2))}
2c()[l
e x
PWv()i))}]
Thus the ratio of the concentration of any two size fractions ct{z) and
C){z) at a given level is given by
ci(z)exp{w(y>(z)} = A(ici(z)eTp{wiy>(z)},
(27)
A(i = ^> exp{(«><-t0/)V(«)}
c,(a)
(28)
where
from (26), or alternatively by ehminating 2 wrcr from the tth and j t h
equations of the system (17).
Consequently,
.
dz
\ct{z)f
e(z)
N
so that the ratio of a fine fraction (c() to a coarser fraction (ct) must
necessarily increase upwards since e > 0, and to, > w{, according to the
simple formula (29).
Batchelor (1) discusses the vertical distribution of a homogeneous
sediment satisfying equation (1), with particular reference to central
regions in channel flow where e —
• > constant, and the constant stress
region near the bed where e <~ z. In both cases, the corresponding
solution of equation (1) gives c(z) -»- oo as z —>- 0 but the ratio w/u^,
where u+ is the usual friction velocity, emerges as a significant parameter.
TURBULENT TRANSPORT OF HETEROGENEOUS SEDIMENT 241
I t is of interest to consider the corresponding solutions to the more
general diffusion equations (19) for a uniform sediment and (17) for a
heterogeneous sediment. The solution to (19) in the case ew = e, was
given in (2) for a uniform sediment with e ~ z(l —z\h).
It will be convenient first to summarise the corresponding results for
a homogeneous sediment.
4. The vertical distribution of a homogeneous sediment
(i) The constant stress region near the lower boundary in a turbulent
channel flow is characterised by a diffusivity
e = Azu*,
(30)
where ut is the friction velocity and A a constant of order unity. The
corresponding solution to equation (19) with e, = ew = e is
c(z) = {l+Kza/Xu'}-\
(31)
where K is a constant. Clearly c(z) —»• 1 as z —*• 0 as was to be expected
from (19). It can be seen that
*
jc{z)dz~z
near z = 0 for all iv/to* $ 1.
(32)
o
This result differs significantly from that found by Batchelor from the
corresponding solution to equation (1), namely
s
c(z) ~ Cz-wn"',
jc(z) dz ~ z 1 -"^".
(33)
o
from which the singularity in c(z) at z = 0 lends a spurious significance
to the value
f
(34)
= 1.
Batchelor concluded that when wjXu^ > 1, most sediment in suspension remained in the constant stress region, while for w/Xu^ < 1, the
bulk of the sediment was to be found above the constant stress region.
No such significance attaches to wlXu^ from the present analysis, since
the inclusion of sediment volume has removed the singularity from c[z).
Indeed, almost the opposite conclusion might be drawn. The distribution
(31) shows that
* = _*» 2 <«««.>-i
'
(35)
dz
ku+
so that the gradient of concentration becomes large within the constant
6092.2—(20 pp.)
B
242
J. N. HUNT
stress region for a light material (tv/Au^ < 1), and not for a heavy material
for which w/Au^ > 1.
(ii) In central regions of the flow where the diffusivity may be treated
as approximately constant, e = e0, the solution to (19) with e, = ew = e0
c[z) = {1+ZX erpttw/co)}-1.
(36)
where Kx is a constant. Just as in the case of the solution to equation (1)
with a constant diffusivity, namely
it is clear that since e0 is proportional to U+, the parameter w\uM is related
to two distinct regimes; almost uniform sediment concentration when
W/M# < 1 and rapidly varying concentration when wju+ > 1.
(iii) Representative of the entire depth of two-dimensional channel flow
is the diffusivity
J
Z
(37)
\
which, substituted into (19), leads to the distribution
C(Z)=\I+KJ-
,
(38)
where Kt is a constant. For all non-zero values of K%, it is clear that
c(0) = 1,
c{h) = 0.
The concentration gradient c'(z) behaves Uke (35) near z = 0 and so
depends critically on the sign of (wjXu^) — 1. The constant Kt is related
to the total sediment in suspension. Table 1 shows values of the total
sediment C, or mean sediment concentration Gjh, where
0
= Jc(z) dz
(39)
for various values of the constants y = w/Xut, and
At the mid-depth, z = \h, the concentration is given by
c(ih) = {1+KJ?}-1,
(40)
which is the same as the function tabulated in Table 1 under the heading
y = 0. Thus for a sediment for which y = wjlu^ = 1, an observed
concentration at mid-depth c = 0-20, immediately indicates that the total
concentration is G = 0-2828A.
TURBULENT TRANSPORT OF HETEROGENEOUS SEDIMENT 243
TABLE 1
Mean Sediment Concentration as a function of the parameters Kz and y
Gjh
I
a
3
4
5
6
7
8
9
IO
13
*4
16
18
3O
«5
30
35
4°
45
50
60
7»
80
90
100
y =o
y = °5
y = i-o
2-0
0-50000
O-33333
0-25000
0-20000
0-16667
0-50000
035518
0-27924
023144
0-19826
0-50000
050000
038629
032396
028280
025295
0-42613
0-38410
0-35518
0-3334O
0-14286
O-I25OO
0-17375
0-15484
O-I3977
0-12746
0-11720
0-23002
0-21170
019664
0-18398
0-17316
0-31608
030179
0-28969
0-27924
0-27008
0-10106
015553
0-14170
0-13050
0-I2I20
0-11334
0-25465
0-24204
0-23144
0-22235
021442
0-09804
0-19826
0-18571
0-17556
0-16710
0-1599°
O'l I I I I
o-ioooo
0-09091
0-07692
0-06667
0-05882
0-05263
0-04762
008892
007943
0-07181
006554
0-03846
0-03226
0-02778
002439
002174
005386
0-04575
0-03979
OO8684
003522
0-07823
OO7137
003159
OO6575
0-01961
0-01639
001408
0-01235
0-01099
0-02865
0-02416
0-02090
0-01841
0-01646
005362
004797
004351
003989
0-15365
0-14330
0-13498
012808
0-12224
0-00990
001488
0-03689
o-i 1720
0-06106
Cjh for y = o is c(\h).
5. The vertical distribution of a heterogeneous sediment
When the diffusivity is given by equation (37), the general solution (26)
for the vertical distribution of concentration cT(z), for the size fraction
of fall velocity wT, becomes
(41)
cr(z) =
where
1-z/h
and
*<«)=
1-a/A
(42)
244
J. N. HUNT
1-0
=10
V,
o-o
1-0
1-0
<HW2) = 10-»
<iW2
= 5
=10-'
V\
>
^
—
1
0-0
—
1
1-0
FIGS. 1 to 4. Sediment Distribution Curves for a mixture of two particle
sizes, with a varying ratio of fall speeds.
TURBULENT TRANSPORT OF HETEROGENEOUS SEDIMENT 246
1-0
e,(A/2)=10-'
Vi
Vt
oo
=5
=10" 1
c(z)
1-0
10
Figs. 1 to 4 (continuod)
246
J. N. HUNT
If we choose the reference level z = a to be the mid-depth z = A/2 for
convenience, then (41) simplifies slightly to
cr(M2)Z
1I6(A/2){1Z-*}'
=
3
Figures 1 to 4 illustrate the behaviour of the solutions (43) for a
sediment consisting of only two size fractions. It can be seen that when
the fall velocities yx and y2 differ by several orders of magnitude, there is
very little overlap in the vertical distributions; the light particles are
found in the upper layers of the flow and the heavy particles near the bed.
When the fall velocities differ only slightly, the distributions overlap to a
considerable extent, as might be expected, but again only the heaviest
material appears in the lowest layer adjacent to the bed.
When the size fractions are ordered so that
w1<wi<
...wr... <wn,
and, similarly
Yi < Yi <--Yr--- <Yn>
it becomes clear from (43) that approaching the bed, as z —> 0,
cr(z)-+0,
r=
1,2,...,TO(44)
I
while approaching the free surface, as z —»- h,
cr(«) —0,
r=l,2,...,n.
(45)
Thus near the bed, only the heaviest size fraction should be found in
significant amounts, on the basis of the present formulation of the problem.
Similar conclusions may be derived from an analysis in which a heterogeneous sediment is supposed to possess a continuous distribution of size
fractions (and fall velocities), rather than the discrete distribution considered here.
REFERENCES
1. G. K. BATOHELOB, Proc. 2nd Australasian Oonf. Hydraulics and Fluid ilech.
(1965) 019-041.
2. J. N. HUNT, Proc. R. Soc. A 224 (1954) 322-335.
3. H. ROUSE, Trans. Am. Soc. civ. Engrs 102 (1937) 463.
4. R. A. BAONOLD, Proc. R. Soc. A 265 (1962) 315-319.
5. A. J. RAUDKTVI, Loose Boundary Hydraulics (Pergamon, 1967) 124.
6. H. M. ISMATL, Trans. Am. Soc. civ. Engrs 117 (1952) 409.
7. S. R. SINOAMSETTI, Proc. Am. Soc. civ. Engrs HY2 (1966) 153.