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EXPERIMENTS ON DENSITY AND TURBIDITY CURRENTS 11. UNIFORM FLOW OF DENSITY CURRENTS
GERARD V. MIDDLETON Department of Geology, McMaster University, Hamilton, Ontario
Received February 12, 1966
ABSTRACT
The basic theory for the average velocity of uniform flow of a density current is now well established. The resistance a t the bottom may be estimated from reasonable assumptions regarding the roughness of the bottom and the size of the current. The principal problem remaining is quantitative estimation of the resistance of the upper (fluid) interface. A review of the literature suggests that this resistance increases with increase in Froude number and decreases with increase in Reynolds number, and the writer's experiments support this hypothesis.
As many turbidity currents are large scale and flow over low slopes of relatively small roughness it seems probable that both the bottom resistance and the resistance a t the upper interface are small.
La theorie pour la vitesse moyenne d'ecoulement uniforme en regime plan est bien demontree. I1 est possible de calculer le coefficient de frottement sur le fond d 'aprb des assomptions raisonables relativement B la rugosite du fond et B la largeur du courant. La probleme principal de reste est l'evaluation quanti- tative du coefficient de frottement de la surface supbrieure du courant. L'examen de la litterature du sujet suggere que ce coefficient de frottement s'augmente avec l'augmentation du nombre de Froude et diminue avec I'augmentation du nombre de Reynolds: les experiences signalees dans ce memoire soutiennent cette hypothbe.
Comme plusieurs des courants de turbidit6 sont d'une grande echelle et comme ils ecoulent sur des fonds de pentes minces et de rugosites legbres, les coefficients de frottement du fond et de la surface supdrieure sont probablement d'un ordre plutBt bas tous les deux.
INTRODUCTION AND REVIEW
This paper is the second in a series of three describing the results of experi- ments performed in a lucite flume, 5 meters long, 50 cm deep, and 15.4 cm wide, a t the W. M. Keck Laboratory, California Institute of Technology. The first paper (Middleton 1966b) described the experimental method and the studies made of the motion of the head of density and turbidity currents. In this second paper the laws of uniform flow of density currents will be discussed with reference to the experiments performed in the 5-m flume with density currents composed of salt water colored with a blue dye. The third paper will describe those aspects of the experiments relating to the deposition of sediment from turbidity currents and the formation of graded beds.
A brief review of the literature on the uniform flow of density currents, and a discussion of the problems of scaling down turbidity currents to laboratory size were given in an earlier paper (Middleton 1966~). In that paper i t was concluded that i t is not possible to study, on a small scale, the uniform flow of turbidity currents bearing coarse sediment in suspension, because i t is
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628 CANADIAN JOURNAL OF EARTH SCIENCES. VOL. 3. 1966
necessary to scale down not only the length dimensions of the current but also the settling velocity of the sediment in suspension. In fact, it still remains dubious whether a turbidity current bearing coarse sediment in suspension ever achieves uniform flow, even a t very large scales. The uncertainty remains because the conditions for "auto-suspension" (Bagnold 1962) are not yet experimentally or theoretically established, so that there is some doubt whether auto-suspensions of sediment coarser than very fine sand take place in nature, and because in nature turbidity currents (other than those observed in lakes and reservoirs) appear to be unsteady phenomena comparable in some respects to snow avalanches, floods caused by dam failure, and similar "surges".
In some cases, however, such as the Grand Banks turbidity current described by Erickson et al. (1952), it appears that turbidity currents have travelled for very great distances across the ocean floor, in which case there must have been a close approximation to uniform flow over a t least part of the flow. For this reason, and because of the relative theoretical simplicity of uniform flow, an attempt must be made to understand the hydraulics of the uniform flow of turbidity currents before one can proceed to a discussion of the mechanisms of sedimentation from non-uniform, unsteady flows.
Most of the theory developed for turbidity current flow does, in fact, refer to steady, uniform flow. Daly (1936), Kuenen (1937, 1952), and others have used a Ch6zy type equation
where u is the velocity of uniform flow, C' is a modified Chkzy coefficient, Ap is the difference in density between the flow and the water above, p is the density of the flow, d is the thickness of the flow, and S is the bottom slope. The general validity of an equation of this type, as applied to density currents of saltwater or fine clay suspensions, was demonstrated by experimental studies by Raynaud (1951), Bata and Bogich (1953), Bata (1959), Blanchet and Villatte (1954), Michon et al. (1955), Bonnefille and Goddet (1959), Levi (1959), f i n Jia-Hua (1960), Levy and Kylyesh (1960), and others.
The problem of predicting the average velocity becomes, therefore, the problem of predicting the variation of the coefficient C'.
In the Ch6zy equation for open channel flow
where R is the hydraulic radius of the channel, C may be written
where fo is the Darcy-Weisbach resistance coefficient. I t is known that for straight rough channels a t high Reynolds numbers, fo is mainly a function of the bed roughness. For smooth channels fo is a function of the Reynolds number (see below, equation 8).
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MIDDLETON : DENSITY CURRENTS
Similarly for density currents we may write
(4) u = Cd(R.9 ,
where
f is no longer the simple Darcy-Weisbach coefficient, since there is resistance not only from the bottom and sides of the channel Cfo) but also from the fluid interface Cf,). For rectangular channels we may write
and
where w is the width of the channel and d is the depth of the uniform under- flow (for example, Fan Jia-Hua 1960).
For hydrodynamically smooth, straight channels, fo may be estimated from the Reynolds number
where v is the kinematic viscosity, by use of a Moody diagram (for example, Hydraulic Institute 1954).
For rough channels, a t high Reynolds numbers, fo does not depend on the Reynolds number but upon the relative roughness. For full-scale turbidity currents fo may be estimated from the behavior of similar types of bottom in rivers. I t appears that the bottoms over which turbidity currents flow are usually composed of mud, which is eroded into a fluted surface by the head of the current. Such a fluted surface should have a bottom resistance close to that of a rippled sand bottom: if a plane bed is formed after deposition of some sediment (as indicated by the lamination in some graded beds) the bottom resistance will decrease somewhat. In either case, f o will depend not only on the size of the roughness elements (flutes, or grains), but also on the thickness of the flow Cfo decreases as d increases). Thus, although it is not easy to predict exactly what fo will be for a given turbidity current, a reasonable approximation is possible.
The resistance a t the fluid interface, measured by the coefficient f l , is much more difficult to predict. By experimental observation it has been established (Keulegan 1944, 1949; Ippen and Harleman 1952; Lofquist 1960; Macagno and Rouse 1962) that the interface may assume a number of different -
physical states. (a) The interface is sharp, with the underflow laminar. (b ) The interface is sharp (because of the presence of a laminar boundary
layer), with the underflow turbulent.
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( c ) There are waves a t the interface which travel downstream: with in- creasing relative motion the waves grow progressively steeper and less regular, and eventually break. Waves may be present a t the interface even where the underflow is laminar.
(d) There is a zone of turbulent mixing a t the interface, with the underflow turbulent.
In (a) where both the flow and the interface boundary layer are laminar a theoretical solution is possible (Ippen and Harleman 1952). A theoretical solution is also possible for the case of a laminar boundary layer with a tur- bulent underflow (Keulegan 1944; Lock 1951 ; Bata 1959). In nature, however, it is most unlikely that the boundary layer is laminar, except for very small, low velocity currents.
The criterion for instability of the interface (that is, the development of waves) was investigated by Keulegan (compare with Harleman 1961), who suggested that the interface was unstable for values of 0 < 0.045, where
where Fr is the densiometric Froude number
As an example, this criterion may be applied to a typical Lake Mead density current, with a velocity of 0.5 ft/s, a thickness of 5 ft, a density of 1.05 (Gould 1951), and a kinematic viscosity of about 2 X ft2/s (Simons, Richardson, and Haushild 1961). The Reynolds number is therefore 5 X lo5 and the Froude number is about 0.18, so that e is about 6 X and the interface must be markedly unstable. This does not necessarily mean, however, that there is extensive mixing.
The degree of mixing across the interface is intimately connected with the resistance a t the interface: the resistance will be larger for mixing than for a laminar boundary layer. The parameter controlling mixing, and therefore the resistance coefficient fi, is not necessarily the same as that controlling stability, although Iwasaki (1964) advanced evidence that the parameter is in fact 8, with fi increasing as 0 increases. This implies that the resistance coefficient varies inversely with the Froude number, which is not in agreement with the results of most other workers.
Ellison and Turner (1959) and Michon, Goddet, and Bonnefille (1955) con- cluded that f depended only on the Froude number and that f increased with increasing Froude number. Lofquist (1960), Macagno and Rouse (1962), and Valembois (1963) concluded that fi depended on both the Froude and Reynolds numbers and that f, increased with increasing Froude number. For constant Froude number, Lofqujst concluded that f i should be proportional to Rc3I5. I t will be shown below that the present writer's experimental results are in agreement with those of Lofquist, and Macagno and Rouse.
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MIDDLETON : DENSITY CURRENTS 631
Because only small ranges of Reynolds number can be examined in the laboratory, and because it is difficult to obtain accurate results from density current experiments, it must be emphasized that it is dangerous to extrapolate these results to large scales. This is particularly important since it is not a t all clear that suspensions behave in exactly the same manner as salt solutions. With these reservations, the general implications of the reported results for turbidity currents are as follows.
(a) The principal control of f, is the Froude number. Provided Fr < 1 there will be relatively little mixing and hence a low resistance coefficient.
In practice this means that there will be little mixing provided that the slope is small (less than 0.5%).
(b) An increase in the size and velocity of the current (that is, in the Reynolds number) will tend to lower the resistance of the upper interface, provided that the slope (and therefore, Fr) remains constant.
(c) The general conclusion is, therefore, that large turbidity currents, flowing over the flat sea floor, will experience relatively little fluid resistance. The resistance a t the bottom and sides will be small because of the large size of the turbidity current (hence high Reynolds number and low relative rough- ness) and the absence of large roughness elements (dunes, sand waves) on the bottom. The resistance a t the upper interface will be even smaller, because of the low Froude number (a consequence of the low slope) and the high Reynolds number. Mixing of the turbidity current with the overlying water is likely to be a slow process, and dissipation of the current will probably result from deceleration of the current (with decreasing slope) and gradual deposition of the sediment, together with processes operating in the head of the current, rather than from dilution across the upper interface.
EXPERIMENTAL RESULTS
The results presented here are based on the analysis of 40 experimental runs, performed with salt solutions in the 5-m tilting flume which was de- scribed in an earlier paper (Middleton 19663). Details of the observations are given in Table I.
The slopes varied from 0.0025 to 0.04 and were measured directly. The solution densities varied from 1.0265 to 1.061 and were measured by means of a specific gravity hydrometer (accurate to about 0.0002). The viscosity of the solution was estimated from the measured density and temperature, by use of data in the International Critical Tables. The discharge of salt water into the flume was maintained constant during the experiment, by use of a constant head tank and adjustable valve. I t was measured by means of an orifice meter and manometer, which was calibrated by weighing the solution discharged into a bucket over a measured period of time. The manometer was checked twice during the running of an experiment. The discharges used varied from about 240 to 900 cc/s. For a single experiment the measurement error and irregular fluctuations in discharge probably did not exceed 20 cc/s.
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CANADIAN JOURNAL OF EARTH SCIENCES. VOL. 3. 1966
TABLE I Results of experiments on uniform flow of saline density currents
Experiment Slope (S) AP d (cm) u (cm/s) v (cmz/s)
Each run was of short duration (1-5 min). Uniform flow-was closely approxi- mated shortly after the passage of the density current head through the flume. The depth reading was made by inspection through the walls of the flume as soon as it became apparent that the readings on several scales attached to the flume a t different distances from the entrance were identical. The measured thickness of uniform flow varied from 0.9 to 4.0 cm. This measure- ment is by far the least accurate of all those made, because of the difficulty of determining accurately the boundary between the flow and the water above. This difficulty was particularly acute a t high slopes, when there was considerable mixing a t the interface. The estimated accuracy is f l O ~ o .
Figure 1 shows the values for f (determined from equations 4 and 5) plotted against Reynolds number. The relationship for smooth pipes is shown for comparison. I t can be seen that, in a general way, the total resistance coefficient, f , decreases with Reynolds number for a given slope, but increases with slope
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MIDDLETON: DENSITY CURRENTS
I
FIG. 1. The total resistance coefficient (f) plotted against Reynolds number (Re = 4uR/v) for several values of slope. Broken lines indicate approximate average trends for slopes of 0.01, 0.02, and 0.04.
(which controls the Froude number). The relationships for f l , calculated from the experimentally determined values for f and the relationship for f, for smooth pipes (equation 7) and averaged for each slope, are shown in Fig. 2. Although there is much scatter, because of experimental error (especially a t large slopes), it is apparent that f l increases with Fr and (to a lesser extent) decreases with Re (the latter is shown only for the more accurate results, a t low slopes).
The importance of the Froude number may also be indicated by plotting a = fi/f, against Fr, as in Fig. 3. If it is assumed that the shear distribution is linear and the flow approximately two-dimensional, then a is also given by a = hl/ho, where hl is the thickness of the flow above the velocity maximum and ho is the thickness of the flow below the velocity maximum (for example,
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CANADIAN JOURNAL OF EARTH SCIENCES. VOL. 3, 1966
FIG. 2. Calculated values for f i plotted against Reynolds number (Re = 4 u R / ~ ) . Points are averages of 2 4 runs.
FIG. 3. fi/fo plotted against Reynolds number for experiments in the 5-m flume. The line indicates the approximate trend, determined by inspection.
HarIeman 1961). The velocity profile was not measured in the experiments reported here, but Michon et al. (1955) reported values of hl and h~ which have been used to construct Fig. 4. I t can be seen that a increases with Froude number in both cases: the velocity maximum lies in the lower half of the flow (a > 1) only for large values of the Froude number (Fr > 2).
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MIDDLETON : DENSITY CURRENTS
FIG. 4. hl/ho, where hl is the thickness of flow above the velocity maximum and ho is the thickness below the velocity maximum. Data from Michon et al. (1955), experiments with clay suspensions in two smooth flumes. The broken line indicates the approximate trend, determined by inspection.
CONCLUSIONS
The experiments reported in this paper are not of a high order of accuracy. In a general way, however, they confirm the results of Lofquist (1960) and others.
(1) The uniform flow of density currents may be represented by an equation of the type
where
and
(2) The value off, may be predicted from diagrams of pipe resistance, if one uses the analogy between density currents and large rivers, and takes into account the type of roughness elements over which most turbidity currents flow.
(3) The value of f may be predicted only in a tentative semiquantitative fashion. For large underflows on low slopes it is probably small, because the
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636 CANADIAN JOURNAL OF EARTH SCIENCES. VOL. 3. 1966
experimental evidence indicates that it decreases with decrease in Froude number and increase in Reynolds number.
(4) A consequence of the relative small value of the ratio fl/f, is that the velocity profile of uniform flow will not show a velocity maximum close to the bottom, but in the upper half of the underflow.
( 5 ) The conclusions stated above may be applied to turbidity currents bearing clay, silt, and very fine sand in suspension. I t is uncertain to what extent they can be applied to currents bearing coarse sediment (sand or gravel).
ACKNOWLEDGMENTS
Financial support which made these experiments possible was provided by Petroleum Research Fund and McMaster University. I am indebted to the staff of the W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, and especially to Mr. Elton Daly and Mr. Robert Greenway for many services, including construction of the apparatus. Drs. Vito Vanoni and Norman Brooks assisted in the design of the apparatus, and gave much helpful discussion and encouragement. Mr. John Buchan acted as research assistant.
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