EARTH SURFACE PROCESSES A N D LANDFORMS, VOL. 7,507-510 (1982)

DISCUSSION OF: EXPERIMENTAL TEST OF AUTOSUSPENSION (SOUTHARD AND MACKINTOSH, 1981)

GARY PARKER St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Mississippi River at Third Avenue, Minneapolis, MN 55414 U.S.A.

Received 27August 1981 Revised 15 June 1982

ABSTRACT

The use of a pipe-flow experiment for testing autosuspension criteria is criticized. It is argued that the proposed criterion for autosuspension is a necessary but not a sufficient condition, and that an analysis of turbidity currents can fi l l this gap.

KEY WORDS Autosuspension Turbidity currents

Bagnold’s elucidation of his concept of self-reinforcing turbidity currents is at once profoundly imaginative and hopelessly muddled. The authors are to be credited for departing from a long line of geomorphologists who have uncritically accepted and applied Bagnold’s ‘autosuspension criterion’.

The writer concurs heartily with the authors’ contention that the ‘autosuspension criterion’

U sin (Y > I

W

does not in itself delineate conditions for which the concentration of suspended sediment can rise indefinitely. The writer also agrees that a key missing element is a description of sediment entrainment and deposition (Parker, 1982) and is gratified to see the authors state this clearly.

It is, however, unfortunate that the authors chose to use pipe flow as a configuration for testing the autosuspension criterion. Their experiments can be combined with a perusal of the equations of motion to demonstrate that inequality (4) has essentially nothing to do with the ultimate concentration attained in pipe or open-channel flows. The result does not generalize to true turbidity currents. Indeed, the experiments are not relevant to this case.

Since the authors have clearly given some thought to the difference in configuration (e.g. p. 107), the above comments need to be carefully justified. Consider a flow in an infinitely wide channel with vertical thickness h . The flow could be pressurized plane pipe flow, open-channel flow, or a turbid density current moving through quiescent clear water (water entrainment is neglected herein for simplicity). The flow may be laden with suspended sediment; concentrations are assumed not to exceed a few per cent, so that, for example, the Boussinesq approximation can be used, and damping of turbulence induced by stratification can be taken to be fairly small. The configurations are shown in Figure 1.

* Equation numbering continues from those used by Southard and Mackintosh (1981).

0197-9337/82/050507-04$01.00 @ 1982 by John Wiley & Sons, Ltd.

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/ I

, I

i ,INTERFACE

I

(d

Figure 1. (a) Plane pipe flow. (b) Open-channel flow. (c) Turbidity current. The parameters u and c denote local mean values of downstream velocity and suspended-sediment concentration; their layer-integrated counterparts are lJ and C

A rather crude layer-integrated energy equation for steady, longitudinally uniform flow is

( 5 ) aP a x 0 0 0 @ Q 8

0 = -uh-+ p (1 - C ) h us + psch us - T b u - Ttu - ( p , - p ) g c h w

These terms are as follows:

1. Rate of energy supply to flow via the mean pressure gradient a P / a x . (Rate of work done on fluid

2. Rate of energy supply to fluid phase via gravity. (Work done on fluid by gravity force.) 3. Rate of energy supply to sediment phase via gravity; C denotes vertically-averaged volumetric

4. Energy loss associated with bed friction Tb.

5 . Energy loss associated with ‘top’ friction T~ (zero for open channel; skin friction for pipe; interfacial friction for turbidity current).

6. Energy expended in suspending sediment. The origin of this term is somewhat more obscure than the others. Since its validity is questioned by the authors, the matter is considered in more detail later. (Note that for simplicity cos (Y has been approximated as unity.)

by pressure force.) Here S =sin (Y is bed slope, assumed to be very small.

concentration.

Equation ( 5 ) is common to all three configurations. The formulation of the pressure term is not. For open-channel flow, @ / a x may be taken to vanish. For pipe flow a P j a x can be equated to the imposed centre line pressure gradient dP,/ax. For turbid underflows in quiescent clear water, however, P(r ) = pg(H - h ) + gj,h [ p + ( p s - p ) c ] dz where ff is the elevation of the water surface above the bed and c (2)

is local mean concentration. For longitudinally uniform flows, then c and h are independent of x , so from the geometry of Figure 1,

Thus energy balance can be cast as follows:

(a) Pipe or open-channel flow (aP, / a x = Tt = 0 for latter)

ap, ax

0 = -Uh-+pUhS + ( p s -p)UChS - (Tb + 7t)U - (p * -p )gChw

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It is apparent from both Equations (6 ) and (7) that more energy is being fed into the sediment phase than is required for suspension if U S / w > 1, i.e. inequality (4). The implications of this fact differ vastly depending on flow configuration. In order to explore them, it is useful to postulate a ‘sediment entrainment’ relation; at equilibrium,

C = f ( ~ h , other parameters) (8)

where f is some empirically-specified function (see Parker, 1981) for a more rigorous formulation). The above relation implies that for a given sediment size and type equilibrium-sediment concentration is strongly dependent on bed stress and only weakly dependent on other parameters such as gradient Richardson number (see e.g. Smith and McLean, 1977).

In Equation (6 ) for pipes or open channels, the two terms involving sediment are usually only modest corrections to the terms involving the fluid phase only. As a result, the equations for flow energetics and suspended-sediment balance decouple neatly. Bed stress q, can be computed from Equation (6) with C = 0 and C can be computed from that value of T b and Equation (8). Successive approximation can be used to reinsert this calculated value of C into Equation (6) , obtain a new (but only slightly different) value of 76, and thus a corrected value of C, but in many cases of interest the correction is likely to be small.

Thus the ‘autosuspension criterion’ in the context of pipe or open-channel flow tells us that one small term in the energy equation is greater than another and nothing else. Neither term contributes much to determining bed stress, and thus ultimate sediment concentration.

It is precisely for this reason that nearly identical discharges and energy slopes yield nearly identical concentrations in the authors’ experiments, regardless of whether or not the autosuspension criterion is satisfied. The experiments clearly indicate that the criterion is meaningless for pipe and open-channel flows.

The case of turbidity currents, i.e. Equation (7), presents a very different picture. Energy enters into the system only via the sediment. If concentration C is equal to zero, no flow can exist because there is no density difference to drive it. Thus, whatever equilibrium flow is realized cannot be said to be energetically ‘only slightly removed’ from some clear water flow. It is apparent from Equation (7) that unless

(ps-p)UChS > (ps-p)gChw

(i.e. inequality (4)), no equilibrium flow whatsoever is possible. It follows that inequality (4) is a necessary condition for a self-sustaining turbidity current. This needs to be emphasized; the authors’ results concerning the meaningless of inequality (4) for pipe and open-channel flows do not carry over to turbidity currents.

On the other hand, the fact that Bagnold termed turbidity currents that satisfy the inequality ‘autosus- pensions’ indicates that he assumed the criterion to be sufficient as well as necessary. Bagnold was in error on this point, for precisely the reasons cited by the authors, namely, failure to consider all of the terms in the energy balance and failure to describe entrainment and deposition.

The writer (Parker, 1982) has recently performed an analysis of turbidity currents in which an attempt is made to take the missing factors into account. The analysis illustrates that inequality (4) is insufficient for self-sustaining turbidity currents. The analysis also indicates the existence of an ‘ignition’ condition for continuous turbidity currents, below which a current dies in time, and above that which the current accelerates toward a state of fairly high concentration and velocity. Many aspects of the treatment are still crude and tentative. However, the existence of an ‘ignition condition’ above and beyond inequality (4) may explain the failure to observe such accelerative currents in the laboratory to date, and may point the way to the design of more appropriate experiments.

A final minor point concerns an addition error in Equation (1) on p. 109. The writer concurs that Bagnold’s derivation for the term denoting energy expended by the flow to suspend sediment, with its illustration of cute little fishes, is little more than articulate handwaving. Nevertheless, it happens to be

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correct. A fairly rigorous derivation can be found in the summary of the work of Barenblatt contained in Bogardi (1974), but beware of many misprints. The authors could also have obtained it correctly from their own simplified calculation. Consider a single column of height h containing fluid and grains in equilibrium suspension. At any instant, C, is the column-averaged concentration of grains that happened to be descending, and C, is the similar concentration of ascending grains; note

(One might assume all the ascending grains to be on the right-hand side and the descending grains to be on the left-hand side, but it is not necessary.) Remembering that the work rate equals the scalar product of the force vector and the vector of particle velocity relative to the fluid, the work rate per unit bottom area is

w = [-bs - p )&ah 1L-W 1 + [ - (ps-p )gCdh I[-w 1 = (ps -p )gchw

since both the force and the relative velocity are always directed downward. Minor errors aside, the authors are to be complimented for this thought-provoking paper.

REFERENCES

Bogardi, 3. 1974. Sediment Transport in Alluuial Streams, Akademiai Kiado, Budapest, Hungary. Southard, J. B. and Mackintosh, Michael, E. 1981. ‘Experimental test of autosuspension’. Earth Surface Processes and Landforms,

Parker, G . 1982. ‘Conditions for the ignition of catastrophically erosive turbidity currents’, Marine Geology, 46, 307-327. Smith, J. D., and McLean, S. R. 1977. ‘Spatially averaged flow over a wavy surface’, Journal of Geophysical Research, 82 (12).

6,103-111.

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