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Sediment entrainment and suspension in a turbulent tidal flow
Marine Geology, 18 (1975) M57--M64 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
Letter Section SEDIMENT ENTRAINMENT AND SUSPENSION IN A T U R B U L E N T TIDAL FLOW C L I F F O R D M. GORDON
Ocean Sciences Division, Naval Research Laboratory, Washington, D.C. (U.S.A.) (Submitted January 1, 1975; accepted February 2, 1975)
ABSTRACT Gordon, C.M., 1975. Sediment entrainment and suspension in a turbulent tidal flow. Mar. Geol., 18: M57--M64. Direct measurements of Reynolds stress and turbulent kinetic energy have been made in a natural, tidal, boundary layer. The results show that for the same mean flow both these quantities increase by at least a factor of two when the longitudinal pressure gradient changes from favorable to adverse. Since b o t t o m stress and turbulent intensity are directly related to sediment movement, decelerating tidal phases would be expected to transport more sediment than accelerating phases. A qualitative description is proposed that relates sediment movement, intermittent high b o t t o m stress and the degree of turbulence through the unifying mechanism of the b u r s t l w e e p cycle.
INTRODUCTION
It is generally accepted that sediment transport in most natural flows is intimately related to the degree of turbulence. Previous measurements by Gordon and Dohne (1973) have shown that the turbulent structure of a tidal boundary layer is influenced by the longitudinal pressure gradient at the particular stage of the tide and is not a function of mean current velocity alone. Since this circumstance could result in a pattern of sediment movement quite different from the case of steady flows, a further investigation has been pursued. This study has concentrated on measuring the tidal-phase dependence of the turbulence parameters directly associated with entrainment and suspension of sediment, i.e., intermittent periods of high, vertical, m o m e n t u m transport, Reynolds stress and turbulent kinetic energy. The experimental results show that each of these quantities varies during the tidal cycle in a way that would enhance sediment movement on decelerating phases of the flow. All the data are consistent with a description of sediment transport in which a major role is played b y the intermittent, burst--sweep cycle in the boundary layer: There is adequate evidence in the literature of the direct relationship between the bursting phenomenon, Reynolds stress, turbulent intensity and
M58 longitudinal pressure gradient to support such a description and predict at least qualitative effects on sediment transport in tidal currents where deceleration of the flow may be attributed to an adverse pressure gradient and vice versa. BACKGROUND The action of turbulent velocity fluctuations in producing intermittent periods of high b o t t o m stress that initiate movement of sediment has been recognized for some time (White, 1940; Kalinske, 1943; Hunt, 1954). The properties of the resultant lift forces at the bed associated with boundarylayer turbulence were measured directly b y Einstein and E1-Samni (1949). Later on the extensive flume observations by Vanoni (1964) established that such forces do indeed produce intermittent bursts of bed-grain movement. He also associated this effect with quasiperiodic disruption of the near-wall, laminar layer as described b y Einstein and Li (1956). With the further development of flow visualization techniques for investigating the turbulent structure of boundary layers (Kline et al., 1967; Corino and Brodkey, 1969) the description of such quasiperiodic disruptions has been greatly refined. The present convention is to refer to these intermittent events collectively as manifestations of the bursting phenomenon or the burst--sweep cycle. The history and present status of this research have been recently reviewed b y Often and Kline (1973). In general the term burst refers to the ejection of a slowly moving parcel of fluid away from the near-wall layer while an inrush of high-speed fluid from the outer boundary layer is called a sweep. The relationship of the bursting phenomenon to intermittent sediment movement has been recognized by Sutherland (1967) and convincingly demonstrated experimentally by Grass (1971). A stochastic model to describe sediment entrainment due to the bursting phenomenon on hydraulically smooth sand beds has been proposed by Blinco et al. {1973). The relatively strong influence of longitudinal pressure gradients on the apparent b o t t o m stress required to initiate bed movement of sand in a flume has been known for some time, due to the observations b y White {1940). Pressure-gradient effects in geophysical flows, particularly on the vertical distribution of stress above b o t t o m , have been measured and theoretically modeled by Bowden et al. (1959) for a tidal current. The current state of research on the problem of sediment transport in flows with shorter period, oscillating, pressure gradients is well documented by Teleki (1972) in his recent study of wave boundary layers. From the point of view of the descriptive model being proposed here, the most significant pressure-gradient effects are the experimental observations (Schraub and Kline, 1965; Kline et al., 1967) that both the frequency and intensity of bursting events in a boundary layer are reduced in favorable gradients and increased when the gradient is adverse to the flow. Therefore, if the intermittent, high, b o t t o m stresses that initiate bed movement are indeed manifestations of the burst--
M59 sweep cycle, this result implies that for a given flow velocity, bed movement and the rate of entrainment of sediment into the flow will be enhanced during those phases of the tide when the current is decelerating. The a m o u n t of suspended sediment in transport is also a function of the turbulent structure of the flow, since the diffusion coefficient for sediment is usually considered to be proportional to the square of the rms turbulent velocity fluctuations or turbulent kinetic energy (Barfield et al., 1969). There is also a growing body of evidence that the bursting phenomenon is the primary mechanism for the generation of boundary-layer turbulence and maintenance of Reynolds stress (Kim et al., 1971; Wallace et al., 1972; Lu and Willmarth, 1973). This at least, implies a causal relationship between suspended-sediment concentration and the burst--sweep cycle. The connection is more intuitively clear when it is recalled that in turbulent shear flows the x--z c o m p o n e n t of the Reynolds stress is a measure of vertical m o m e n t u m transport and to a first approximation is directly proportional to the vertical transport of material in suspension, assuming a concentration gradient similar to the mean velocity gradient. The linear variation of turbulent kinetic energy with Reynolds stress in shear flows has been established as an empirical fact (Harsha and Lee, 1970) and proved to be a sound basis for theoretical boundary-layer models (Bradshaw et al., 1967). Thus a tidal pressure gradient adverse to the flow would be expected to increase the a m o u n t of sediment transported in suspension for a given mean current. In summary, it is clear from a survey of the literature that there are adequate grounds for postulating a direct causal relationship between the longitudinal pressure gradient in a tide and the flow variables responsible for the transport of non-cohesive sediment. The burst--sweep cycle in the boundary layer suggests itself as the underlying physical process and unifying concept. A qualitative description based on the evidence presented would predict more bursting events, higher Reynolds stresses, greater turbulent kinetic energy and more sediment transport during tidal phases with decelerating currents. It is implicitly assumed that deceleration of the flow may be attributed to an adverse pressure gradient and that b o t t o m stress is roughly proportional to the Reynolds stress. METHOD In order to obtain field data on the tidal-phase dependence of the bursting rate, Reynolds stress and turbulent kinetic energy a pivoted-vane current meter was suspended in the boundary layer of an estuarine tidal flow. The depth of the velocity sensor was 5.75 m. The b o t t o m was about 8 m below the surface. This depth z corresponds to a location z / 6 ~ 0.5 relative to the boundary layer thickness 6 obtained from a mean velocity profile. According to Lu and Willmarth (1973) the bursting rate is not particularly sensitive to z / 6 within the range from 0.10 to 0.70 so this depth should provide representative measurements. Time series of horizontal and vertical mean current velocities ( U, W),
M60
-~
2.25 METERS OFF BOTTOM FLOOD TiDE CURRENT (o) AND STRESS (a)
05
I0
15
2.0
25
30
35
40
45
50
55
60
T I M E iN HOURS
Fig.1. Current speed and Reynolds stress during 1/2 tidal cycle showing the stress lagging the current. The dark points indicate intervals analyzed for pressure-gradient effects.
20 0 ACCELERATING
FLOOD TIDE
DECELERATING
FLOOD TIDE
.! E o E t~
AA
16 0
LJ 12 ,~.
O/0
~s 0
O+/o
4
©
o!
,0
,
©
20
3'0 CURRENT
4'0 SPEED
50 60 (cm/sec)
70
80
Fig.2. The "hysteresis" of the Reynolds stress as a function of current speed. r
5 G"
0
i
i
--
[
I
i
ACCELERATING FLOOD TIDE
A
Z& DECELERATING FLOOD TIDE /
oo
I ]
v
uJ 3 rr
0
o
f_ ~.... A
aa~o [ I0
i0 20
50
~ 40
o . - --~
o
50
i 60
o i 70
80
CURRENT SPEED (cm/sec)
Fig.3. The "hysteresis" of the turbulent kinetic energy as a function of current speed.
M61
their t u r b u l e n t f l u c t u a t i o n s (u,w) and the p r o d u c t u w were r e c o r d e d for 6 hours during a flood tide. T h e sampling interval was 2 sec. T h e details o f the field site where the m e a s u r e m e n t s were made ( C h o p t a n k River), the o p e r a t i o n of the pivoted-vane m e t e r and the data t r e a t m e n t t e c h n i q u e s have been described previously {Gordon and D o h n e , 1973) and will not be r e p e a t e d here. It should be n o t e d , however, t h a t calculations based o n tidetable data show a fairly linear relationship b e t w e e n dP/dx and the t e m p o r a l acceleration o f the mean c u r r e n t ( d U / d t ) at this e x p e r i m e n t a l site. T h e measured c u r r e n t speeds and R e y n o l d s stresses averaged over 9-rain intervals are s h o w n in Fig. 1. T h e stresses were o b t a i n e d b y the e d d y correlation m e t h o d , t h a t is, the vertical R e y n o l d s stress c o m p o n e n t 7xz = - - p u w . It is clear f r o m the figure t h a t the R e y n o l d s stress m a x i m u m lags the current velocity by a p p r o x i m a t e l y an hour. If the b o t t o m stress (To) follows the same p a t t e r n the c o n v e n t i o n a l m e t h o d for d e t e r m i n i n g its value, ~o = kU~, m a y lead to errors when applied to tidal flows. This n o n - s y m m e t r i c a l d e p e n d e n c e o f R e y n o l d s stress on c u r r e n t speed is seen even m o r e graphically in Fig. 2. Since t u r b u l e n t kinetic energy varies linearly with R e y n o l d s stress is shear flows it would be e x p e c t e d to exhibit a similar pattern. This is conf i r m e d by Fig. 3. T h e kinetic-energy values {pq~/2 where q2 = u2 + v 2 + w 2 ) p l o t t e d in Fig. 3 are s o m e w h a t a r b i t r a r y since o n l y u 2 and w 2 were measured directly. F o r purposes of this graph v 2 was t a k e n to be 0.5 u 2 . T h e significant aspect o f b o t h figures is t h a t t u r b u l e n t kinetic energy and R e y n o l d s stress are c o n s i s t e n t l y larger on the decelerating phase of the tidal flow, that is, in qualitative a g r e e m e n t with the p r o p o s e d model. In order to f u r t h e r isolate the influence of the pressure gradient, the two 45-min intervals indicated by the d a r k e n e d points in Fig. 1 were investigated in greater detail. During these two time periods a p p r o x i m a t e l y the same mean c u r r e n t was flowing in the same d i r e c t i o n over the same b o t t o m . T h e r e f o r e it will be assumed t h a t external c o n d i t i o n s during these two intervals vary
TABLE 1 The influence of pressure gradient on factors controlling sediment movement Measurement
dP/dx, F a v o r a b l e
dP/dx, A d v e r s e
Average current g cm sec Acceleration dU/dt c m s e c - : Burst-sweep events !puwl > 15 d y n e c m 2 Reynolds stress --p~ dyne cm-2 Turbulent kinetic energy g m cm 2 see -2
51.5
51.7
+9.10- 3
--6.10- 3
10
51
1.0
2.9
5.2
11.2
M62
only in longitudinal pressure gradients (dP/dx) and hence any measured differences in their turbulent structure may be considered an effect due to this variable alone. The results of this comparison are summarized in Table I. The number of burst--sweep events shown in the table was obtained by a computer search of the time series of u w products for intermittent periods of high m o m e n t u m transport. Selection criteria for bursts (ejections) were u < 0, w > 0 and ]uwl > 15; for sweeps (inrushes) u > 0, w < 0 and luwl > 15. It is seen from the table that all the variables related to sediment transport are 2--5 times greater in magnitude when dP/dx is adverse to the flow. CONCLUSIONS
Even though the current sensor is " b a n d w i d t h " limited to measurements of eddy dimensions of approximately 1 m or larger, the presence of intermittent periods of high m o m e n t u m transport in the tidal boundary layer is clearly demonstrated. Both Heathershaw (1974) and Gordon (1974) have previously reported "events" of this kind and independently attributed them to manifestations of the bursting phenomenon on a geophysical scale. Evidence in support of this interpretation includes the fact that the events contribute nearly 100% of the Reynolds stress. This is characteristic of the burst--sweep cycle (Lu and Willmarth, 1973; Wallace et al., 1972). Furthermore, based on the results of Rao et al. (1971), the rate of occurrence of such events appears to be within the range expected for high-Reynolds-number, boundary layers. Their survey of the available experimental data indicates that the period between bursts T ~ k5 */U, where 5 * is the displacement thickness and k varies between 30 and 70 depending on the somewhat arbitrary definition of a "burst". The theoretical basis for anticipating large-scale intermittence and discussions of its geophysical significance can be found in the work of MolloChristensen (1973) and Grass (1971). It should be pointed out, however, that the identification of this kind of large-scale intermittent phenomena with the small-scale bursts described by Kline et al. (1967) is open to question (Antonia, 1972). The apparent effects of the longitudinal pressure gradient of the tidal flow on the bursting rate, the resultant Reynolds stress and the turbulent kinetic energy are readily measurable and are in qualitative agreement with the description proposed. However, few experimental measurements of this t y p e have been published and consequently few confirming data are available. Among the field experiments that have yielded results consistent with the present interpretation are those of McCave (1973), Kachel and Sternberg (1971) and Thorn (1974). Some of the shear velocities u * ( ~ / 7 o ) derived by McCave (1973) from velocity profiles in an analogous, tidal, boundary layer do exhibit very similar "hysteresis" with respect to the mean current as that shown in Fig. 2. Kachel and Sternberg (1971) have observed an apparent increase in bedload transport as ripples during the decelerating phase of an ebb tide. Thorn (1974) has reported "hysteresis" of fine sand
M63
suspensions in a tidal estuary with respect to accelerating and decelerating currents. Unfortunately, in none of these cases can the phase dependence be unambiguously attributed to pressure-gradient effects. In conclusion, a somewhat circuitous b u t quite plausible relationship has been established between the pressure gradient, the bursting phenomenon, Reynolds stress, turbulent kinetic energy and sediment transport in a boundary layer. The influence of the longitudinal pressure gradient in a tidal flow is such that the factors affecting sediment movement are enhanced during decelerating phases. If the interpretation of intermittent periods of high m o m e n t u m transport as manifestations of the bursting phenomenon is correct, then the measurements presented here support the qualitative model proposed. This description predicts that for the same mean current the amount of sediment in transport on decelerating tidal phases will be greater than on accelerating phases. ACKNOWLEDGEMENTS
The author wishes to acknowledge the invaluable contribution of Mr. C.F. Dohne who was largely responsible for the design and maintenance of the instrumentation used in this work. Helpful discussions with Dr. I.N. McCave, Dr. G. Gust and Dr. D. Polis are also gratefully acknowledged. REFERENCES Antonia, R.A., 1972. Conditionally sampled measurements near the outer edge of a turbulent boundary layer. J. Fluid Mech., 56: 1--18. Barfield, B.J., Smerdon, E.T. and Hiler, E.A., 1969. Prediction of sediment profiles in open channel flow by turbulent diffusion theory. Water Resour. Res., 5: 291--299. Blinco, P.H., Mahmood, K. and Simons, D.B., 1973. Stochastic structure of the turbulent boundary shear stress process. Proc. 15th Congr. IAHR, 1: A47-1--A47-10. Bowden, K.F., Fairbairn, L.A. and Hughes, P., 1959. The distribution of shearing stresses in a tidal current. Geophys. J. R. Astron. Soc., 2: 288--305. Bradshaw, P., Ferriss, D.H. and Atwell, N.P., 1967. Calculation of boundary-layer development using the turbulent energy equation. J. Fluid Mech., 28: 593--616. Corino, E.R. and Brodkey, R.S., 1969. A visual investigation of the wall region in turbulent flow. J. Fluid Mech., 37: 1--30. Einstein, H.W. and E1-Samni, E.A., 1949. Hydrodynamic forces on a rough wall. Rev. Mod. Phys., 21: 520--524. Einstein, H.A. and Li, H., 1956. The viscous sublayer along a smooth boundary. J. Eng. Mech. Div. Am. Soc. Cir. Eng., 82: 945-1--945-27. Gordon, C.M., 1974. Intermittent m o m e n t u m transport in a geophysical boundary layer. Nature, 248: 393--394. Gordon, C.M. and Dohne, C.F., 1973. Some observations of turbulent flow in a tidal estuary. J. Geophys. Res., 78: 1971--1978. Grass, A.J., 1971. Structural features of turbulent flow over smooth and rough boundaries. J. Fluid Mech., 50: 233--255. Harsha, P.T. and Lee, S.C., 1970. Correlation between turbulent shear stress and turbulent kinetic energy. AIAA J., 8: 1508--1510. Heathershaw, A.D., 1974. "Bursting" phenomena in the sea. Nature, 248: 394--395.
M64
Hunt, J.N., 1954. The turbulent transport of suspended sediment in open channels. Proc. R. Soc. Lond., 224: 322--335. Kachel, N.B. and Sternberg, R.W., 1971. Transport of bedload as ripples during an ebb current. Mar. Geol., 10: 229--244. Kalinske, A.A., 1943. Turbulence and the transport of sand and siltby wind. Ann. N.Y. Acad. Sci., 44: 41--54. Kim, H.T., Kline, S.J. and Reynolds, W.C., 1971. The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech., 50: 133--160. Kline, S.J., Reynolds, W.C., Schraub, F.A. and Rundstadler, P.W., 1967. The structure of turbulent boundary layers. J. Fluid Mech., 30: 741--773. Lu, S.S. and Willmarth, W.W., 1973. Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J. Fluid Mech., 60: 481--511. McCave, I.N., 1973. Some boundary-layer characteristics of tidal currents bearing sand in suspension. Mere. Soc. R. Sci., 6th Series, Liege, 6: 107--126. Mollo-Christensen, E., 1973. Intermittence in large-scale turbulent flows. Ann. Rev. Fluid Mech., 5: 101--118. Offen, G.R. and Kline, S.J., 1973. Experiments on the velocity characteristics of "bursts" and on the interactions between the inner and outer regions of a turbulent boundary layer. Stanford Univ. Rep., MD-31:230 pp. Rao, K.N., Narasimha, R. and Badri Narayanan, M.A., 1971. The "bursting" phenomenon in a turbulent boundary layer. J. Fluid Mech., 48: 339--352. Schraub, F.A. and Kline, S.J., 1965. A study of the structure of the turbulent boundary layer with and without longitudinal pressure gradients. Stanford Univ. Rep., MD-12: 157 pp. Sutherland, A.J., 1967. Proposed mechanism for sediment entrainment by turbulent flows. J. Geophys. Res., 72: 6183--6194. Teleki, P.G., 1972. Wave boundary layers and their relation to sediment transport. In: D.J.P. Swift, D.B. Duane and O.H. Pilkey (Editors), Shelf Sediment Transport, Process and Pattern. Dowden, Hutchinson Ross, Stroudsburg, Pa., pp. 21--59. Thorn, M.F.C., 1974. Hysteresis of fine sand suspensions in a tidal estuary. Hydraulics Res. Sta. Note No. 17:3 pp. Vanoni, V.A., 1964. Measurements of critical shear stress for entraining fine sediments in a boundary layer. Calif. Inst. Technol. Rep. KH-R, 7. Wallace, J.M., Eckelmann, H. and Brodkey, R.S., 1972. The wall region in turbulent shear flow. J. Fluid Mech., 54: 39--48. White, C.M., 1940. The equilibrium of grains on the bed of a stream. Proc. R. Soc. Lond., 174: 322--338. Willmarth, W.W. and Lu, S.S., 1972. The structure of Reynolds stress near the wall. J. Fluid Mech., 55: 65--92;
Letter Section SEDIMENT ENTRAINMENT AND SUSPENSION IN A T U R B U L E N T TIDAL FLOW C L I F F O R D M. GORDON
Ocean Sciences Division, Naval Research Laboratory, Washington, D.C. (U.S.A.) (Submitted January 1, 1975; accepted February 2, 1975)
ABSTRACT Gordon, C.M., 1975. Sediment entrainment and suspension in a turbulent tidal flow. Mar. Geol., 18: M57--M64. Direct measurements of Reynolds stress and turbulent kinetic energy have been made in a natural, tidal, boundary layer. The results show that for the same mean flow both these quantities increase by at least a factor of two when the longitudinal pressure gradient changes from favorable to adverse. Since b o t t o m stress and turbulent intensity are directly related to sediment movement, decelerating tidal phases would be expected to transport more sediment than accelerating phases. A qualitative description is proposed that relates sediment movement, intermittent high b o t t o m stress and the degree of turbulence through the unifying mechanism of the b u r s t l w e e p cycle.
INTRODUCTION
It is generally accepted that sediment transport in most natural flows is intimately related to the degree of turbulence. Previous measurements by Gordon and Dohne (1973) have shown that the turbulent structure of a tidal boundary layer is influenced by the longitudinal pressure gradient at the particular stage of the tide and is not a function of mean current velocity alone. Since this circumstance could result in a pattern of sediment movement quite different from the case of steady flows, a further investigation has been pursued. This study has concentrated on measuring the tidal-phase dependence of the turbulence parameters directly associated with entrainment and suspension of sediment, i.e., intermittent periods of high, vertical, m o m e n t u m transport, Reynolds stress and turbulent kinetic energy. The experimental results show that each of these quantities varies during the tidal cycle in a way that would enhance sediment movement on decelerating phases of the flow. All the data are consistent with a description of sediment transport in which a major role is played b y the intermittent, burst--sweep cycle in the boundary layer: There is adequate evidence in the literature of the direct relationship between the bursting phenomenon, Reynolds stress, turbulent intensity and
M58 longitudinal pressure gradient to support such a description and predict at least qualitative effects on sediment transport in tidal currents where deceleration of the flow may be attributed to an adverse pressure gradient and vice versa. BACKGROUND The action of turbulent velocity fluctuations in producing intermittent periods of high b o t t o m stress that initiate movement of sediment has been recognized for some time (White, 1940; Kalinske, 1943; Hunt, 1954). The properties of the resultant lift forces at the bed associated with boundarylayer turbulence were measured directly b y Einstein and E1-Samni (1949). Later on the extensive flume observations by Vanoni (1964) established that such forces do indeed produce intermittent bursts of bed-grain movement. He also associated this effect with quasiperiodic disruption of the near-wall, laminar layer as described b y Einstein and Li (1956). With the further development of flow visualization techniques for investigating the turbulent structure of boundary layers (Kline et al., 1967; Corino and Brodkey, 1969) the description of such quasiperiodic disruptions has been greatly refined. The present convention is to refer to these intermittent events collectively as manifestations of the bursting phenomenon or the burst--sweep cycle. The history and present status of this research have been recently reviewed b y Often and Kline (1973). In general the term burst refers to the ejection of a slowly moving parcel of fluid away from the near-wall layer while an inrush of high-speed fluid from the outer boundary layer is called a sweep. The relationship of the bursting phenomenon to intermittent sediment movement has been recognized by Sutherland (1967) and convincingly demonstrated experimentally by Grass (1971). A stochastic model to describe sediment entrainment due to the bursting phenomenon on hydraulically smooth sand beds has been proposed by Blinco et al. {1973). The relatively strong influence of longitudinal pressure gradients on the apparent b o t t o m stress required to initiate bed movement of sand in a flume has been known for some time, due to the observations b y White {1940). Pressure-gradient effects in geophysical flows, particularly on the vertical distribution of stress above b o t t o m , have been measured and theoretically modeled by Bowden et al. (1959) for a tidal current. The current state of research on the problem of sediment transport in flows with shorter period, oscillating, pressure gradients is well documented by Teleki (1972) in his recent study of wave boundary layers. From the point of view of the descriptive model being proposed here, the most significant pressure-gradient effects are the experimental observations (Schraub and Kline, 1965; Kline et al., 1967) that both the frequency and intensity of bursting events in a boundary layer are reduced in favorable gradients and increased when the gradient is adverse to the flow. Therefore, if the intermittent, high, b o t t o m stresses that initiate bed movement are indeed manifestations of the burst--
M59 sweep cycle, this result implies that for a given flow velocity, bed movement and the rate of entrainment of sediment into the flow will be enhanced during those phases of the tide when the current is decelerating. The a m o u n t of suspended sediment in transport is also a function of the turbulent structure of the flow, since the diffusion coefficient for sediment is usually considered to be proportional to the square of the rms turbulent velocity fluctuations or turbulent kinetic energy (Barfield et al., 1969). There is also a growing body of evidence that the bursting phenomenon is the primary mechanism for the generation of boundary-layer turbulence and maintenance of Reynolds stress (Kim et al., 1971; Wallace et al., 1972; Lu and Willmarth, 1973). This at least, implies a causal relationship between suspended-sediment concentration and the burst--sweep cycle. The connection is more intuitively clear when it is recalled that in turbulent shear flows the x--z c o m p o n e n t of the Reynolds stress is a measure of vertical m o m e n t u m transport and to a first approximation is directly proportional to the vertical transport of material in suspension, assuming a concentration gradient similar to the mean velocity gradient. The linear variation of turbulent kinetic energy with Reynolds stress in shear flows has been established as an empirical fact (Harsha and Lee, 1970) and proved to be a sound basis for theoretical boundary-layer models (Bradshaw et al., 1967). Thus a tidal pressure gradient adverse to the flow would be expected to increase the a m o u n t of sediment transported in suspension for a given mean current. In summary, it is clear from a survey of the literature that there are adequate grounds for postulating a direct causal relationship between the longitudinal pressure gradient in a tide and the flow variables responsible for the transport of non-cohesive sediment. The burst--sweep cycle in the boundary layer suggests itself as the underlying physical process and unifying concept. A qualitative description based on the evidence presented would predict more bursting events, higher Reynolds stresses, greater turbulent kinetic energy and more sediment transport during tidal phases with decelerating currents. It is implicitly assumed that deceleration of the flow may be attributed to an adverse pressure gradient and that b o t t o m stress is roughly proportional to the Reynolds stress. METHOD In order to obtain field data on the tidal-phase dependence of the bursting rate, Reynolds stress and turbulent kinetic energy a pivoted-vane current meter was suspended in the boundary layer of an estuarine tidal flow. The depth of the velocity sensor was 5.75 m. The b o t t o m was about 8 m below the surface. This depth z corresponds to a location z / 6 ~ 0.5 relative to the boundary layer thickness 6 obtained from a mean velocity profile. According to Lu and Willmarth (1973) the bursting rate is not particularly sensitive to z / 6 within the range from 0.10 to 0.70 so this depth should provide representative measurements. Time series of horizontal and vertical mean current velocities ( U, W),
M60
-~
2.25 METERS OFF BOTTOM FLOOD TiDE CURRENT (o) AND STRESS (a)
05
I0
15
2.0
25
30
35
40
45
50
55
60
T I M E iN HOURS
Fig.1. Current speed and Reynolds stress during 1/2 tidal cycle showing the stress lagging the current. The dark points indicate intervals analyzed for pressure-gradient effects.
20 0 ACCELERATING
FLOOD TIDE
DECELERATING
FLOOD TIDE
.! E o E t~
AA
16 0
LJ 12 ,~.
O/0
~s 0
O+/o
4
©
o!
,0
,
©
20
3'0 CURRENT
4'0 SPEED
50 60 (cm/sec)
70
80
Fig.2. The "hysteresis" of the Reynolds stress as a function of current speed. r
5 G"
0
i
i
--
[
I
i
ACCELERATING FLOOD TIDE
A
Z& DECELERATING FLOOD TIDE /
oo
I ]
v
uJ 3 rr
0
o
f_ ~.... A
aa~o [ I0
i0 20
50
~ 40
o . - --~
o
50
i 60
o i 70
80
CURRENT SPEED (cm/sec)
Fig.3. The "hysteresis" of the turbulent kinetic energy as a function of current speed.
M61
their t u r b u l e n t f l u c t u a t i o n s (u,w) and the p r o d u c t u w were r e c o r d e d for 6 hours during a flood tide. T h e sampling interval was 2 sec. T h e details o f the field site where the m e a s u r e m e n t s were made ( C h o p t a n k River), the o p e r a t i o n of the pivoted-vane m e t e r and the data t r e a t m e n t t e c h n i q u e s have been described previously {Gordon and D o h n e , 1973) and will not be r e p e a t e d here. It should be n o t e d , however, t h a t calculations based o n tidetable data show a fairly linear relationship b e t w e e n dP/dx and the t e m p o r a l acceleration o f the mean c u r r e n t ( d U / d t ) at this e x p e r i m e n t a l site. T h e measured c u r r e n t speeds and R e y n o l d s stresses averaged over 9-rain intervals are s h o w n in Fig. 1. T h e stresses were o b t a i n e d b y the e d d y correlation m e t h o d , t h a t is, the vertical R e y n o l d s stress c o m p o n e n t 7xz = - - p u w . It is clear f r o m the figure t h a t the R e y n o l d s stress m a x i m u m lags the current velocity by a p p r o x i m a t e l y an hour. If the b o t t o m stress (To) follows the same p a t t e r n the c o n v e n t i o n a l m e t h o d for d e t e r m i n i n g its value, ~o = kU~, m a y lead to errors when applied to tidal flows. This n o n - s y m m e t r i c a l d e p e n d e n c e o f R e y n o l d s stress on c u r r e n t speed is seen even m o r e graphically in Fig. 2. Since t u r b u l e n t kinetic energy varies linearly with R e y n o l d s stress is shear flows it would be e x p e c t e d to exhibit a similar pattern. This is conf i r m e d by Fig. 3. T h e kinetic-energy values {pq~/2 where q2 = u2 + v 2 + w 2 ) p l o t t e d in Fig. 3 are s o m e w h a t a r b i t r a r y since o n l y u 2 and w 2 were measured directly. F o r purposes of this graph v 2 was t a k e n to be 0.5 u 2 . T h e significant aspect o f b o t h figures is t h a t t u r b u l e n t kinetic energy and R e y n o l d s stress are c o n s i s t e n t l y larger on the decelerating phase of the tidal flow, that is, in qualitative a g r e e m e n t with the p r o p o s e d model. In order to f u r t h e r isolate the influence of the pressure gradient, the two 45-min intervals indicated by the d a r k e n e d points in Fig. 1 were investigated in greater detail. During these two time periods a p p r o x i m a t e l y the same mean c u r r e n t was flowing in the same d i r e c t i o n over the same b o t t o m . T h e r e f o r e it will be assumed t h a t external c o n d i t i o n s during these two intervals vary
TABLE 1 The influence of pressure gradient on factors controlling sediment movement Measurement
dP/dx, F a v o r a b l e
dP/dx, A d v e r s e
Average current g cm sec Acceleration dU/dt c m s e c - : Burst-sweep events !puwl > 15 d y n e c m 2 Reynolds stress --p~ dyne cm-2 Turbulent kinetic energy g m cm 2 see -2
51.5
51.7
+9.10- 3
--6.10- 3
10
51
1.0
2.9
5.2
11.2
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only in longitudinal pressure gradients (dP/dx) and hence any measured differences in their turbulent structure may be considered an effect due to this variable alone. The results of this comparison are summarized in Table I. The number of burst--sweep events shown in the table was obtained by a computer search of the time series of u w products for intermittent periods of high m o m e n t u m transport. Selection criteria for bursts (ejections) were u < 0, w > 0 and ]uwl > 15; for sweeps (inrushes) u > 0, w < 0 and luwl > 15. It is seen from the table that all the variables related to sediment transport are 2--5 times greater in magnitude when dP/dx is adverse to the flow. CONCLUSIONS
Even though the current sensor is " b a n d w i d t h " limited to measurements of eddy dimensions of approximately 1 m or larger, the presence of intermittent periods of high m o m e n t u m transport in the tidal boundary layer is clearly demonstrated. Both Heathershaw (1974) and Gordon (1974) have previously reported "events" of this kind and independently attributed them to manifestations of the bursting phenomenon on a geophysical scale. Evidence in support of this interpretation includes the fact that the events contribute nearly 100% of the Reynolds stress. This is characteristic of the burst--sweep cycle (Lu and Willmarth, 1973; Wallace et al., 1972). Furthermore, based on the results of Rao et al. (1971), the rate of occurrence of such events appears to be within the range expected for high-Reynolds-number, boundary layers. Their survey of the available experimental data indicates that the period between bursts T ~ k5 */U, where 5 * is the displacement thickness and k varies between 30 and 70 depending on the somewhat arbitrary definition of a "burst". The theoretical basis for anticipating large-scale intermittence and discussions of its geophysical significance can be found in the work of MolloChristensen (1973) and Grass (1971). It should be pointed out, however, that the identification of this kind of large-scale intermittent phenomena with the small-scale bursts described by Kline et al. (1967) is open to question (Antonia, 1972). The apparent effects of the longitudinal pressure gradient of the tidal flow on the bursting rate, the resultant Reynolds stress and the turbulent kinetic energy are readily measurable and are in qualitative agreement with the description proposed. However, few experimental measurements of this t y p e have been published and consequently few confirming data are available. Among the field experiments that have yielded results consistent with the present interpretation are those of McCave (1973), Kachel and Sternberg (1971) and Thorn (1974). Some of the shear velocities u * ( ~ / 7 o ) derived by McCave (1973) from velocity profiles in an analogous, tidal, boundary layer do exhibit very similar "hysteresis" with respect to the mean current as that shown in Fig. 2. Kachel and Sternberg (1971) have observed an apparent increase in bedload transport as ripples during the decelerating phase of an ebb tide. Thorn (1974) has reported "hysteresis" of fine sand
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suspensions in a tidal estuary with respect to accelerating and decelerating currents. Unfortunately, in none of these cases can the phase dependence be unambiguously attributed to pressure-gradient effects. In conclusion, a somewhat circuitous b u t quite plausible relationship has been established between the pressure gradient, the bursting phenomenon, Reynolds stress, turbulent kinetic energy and sediment transport in a boundary layer. The influence of the longitudinal pressure gradient in a tidal flow is such that the factors affecting sediment movement are enhanced during decelerating phases. If the interpretation of intermittent periods of high m o m e n t u m transport as manifestations of the bursting phenomenon is correct, then the measurements presented here support the qualitative model proposed. This description predicts that for the same mean current the amount of sediment in transport on decelerating tidal phases will be greater than on accelerating phases. ACKNOWLEDGEMENTS
The author wishes to acknowledge the invaluable contribution of Mr. C.F. Dohne who was largely responsible for the design and maintenance of the instrumentation used in this work. Helpful discussions with Dr. I.N. McCave, Dr. G. Gust and Dr. D. Polis are also gratefully acknowledged. REFERENCES Antonia, R.A., 1972. Conditionally sampled measurements near the outer edge of a turbulent boundary layer. J. Fluid Mech., 56: 1--18. Barfield, B.J., Smerdon, E.T. and Hiler, E.A., 1969. Prediction of sediment profiles in open channel flow by turbulent diffusion theory. Water Resour. Res., 5: 291--299. Blinco, P.H., Mahmood, K. and Simons, D.B., 1973. Stochastic structure of the turbulent boundary shear stress process. Proc. 15th Congr. IAHR, 1: A47-1--A47-10. Bowden, K.F., Fairbairn, L.A. and Hughes, P., 1959. The distribution of shearing stresses in a tidal current. Geophys. J. R. Astron. Soc., 2: 288--305. Bradshaw, P., Ferriss, D.H. and Atwell, N.P., 1967. Calculation of boundary-layer development using the turbulent energy equation. J. Fluid Mech., 28: 593--616. Corino, E.R. and Brodkey, R.S., 1969. A visual investigation of the wall region in turbulent flow. J. Fluid Mech., 37: 1--30. Einstein, H.W. and E1-Samni, E.A., 1949. Hydrodynamic forces on a rough wall. Rev. Mod. Phys., 21: 520--524. Einstein, H.A. and Li, H., 1956. The viscous sublayer along a smooth boundary. J. Eng. Mech. Div. Am. Soc. Cir. Eng., 82: 945-1--945-27. Gordon, C.M., 1974. Intermittent m o m e n t u m transport in a geophysical boundary layer. Nature, 248: 393--394. Gordon, C.M. and Dohne, C.F., 1973. Some observations of turbulent flow in a tidal estuary. J. Geophys. Res., 78: 1971--1978. Grass, A.J., 1971. Structural features of turbulent flow over smooth and rough boundaries. J. Fluid Mech., 50: 233--255. Harsha, P.T. and Lee, S.C., 1970. Correlation between turbulent shear stress and turbulent kinetic energy. AIAA J., 8: 1508--1510. Heathershaw, A.D., 1974. "Bursting" phenomena in the sea. Nature, 248: 394--395.
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Hunt, J.N., 1954. The turbulent transport of suspended sediment in open channels. Proc. R. Soc. Lond., 224: 322--335. Kachel, N.B. and Sternberg, R.W., 1971. Transport of bedload as ripples during an ebb current. Mar. Geol., 10: 229--244. Kalinske, A.A., 1943. Turbulence and the transport of sand and siltby wind. Ann. N.Y. Acad. Sci., 44: 41--54. Kim, H.T., Kline, S.J. and Reynolds, W.C., 1971. The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech., 50: 133--160. Kline, S.J., Reynolds, W.C., Schraub, F.A. and Rundstadler, P.W., 1967. The structure of turbulent boundary layers. J. Fluid Mech., 30: 741--773. Lu, S.S. and Willmarth, W.W., 1973. Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J. Fluid Mech., 60: 481--511. McCave, I.N., 1973. Some boundary-layer characteristics of tidal currents bearing sand in suspension. Mere. Soc. R. Sci., 6th Series, Liege, 6: 107--126. Mollo-Christensen, E., 1973. Intermittence in large-scale turbulent flows. Ann. Rev. Fluid Mech., 5: 101--118. Offen, G.R. and Kline, S.J., 1973. Experiments on the velocity characteristics of "bursts" and on the interactions between the inner and outer regions of a turbulent boundary layer. Stanford Univ. Rep., MD-31:230 pp. Rao, K.N., Narasimha, R. and Badri Narayanan, M.A., 1971. The "bursting" phenomenon in a turbulent boundary layer. J. Fluid Mech., 48: 339--352. Schraub, F.A. and Kline, S.J., 1965. A study of the structure of the turbulent boundary layer with and without longitudinal pressure gradients. Stanford Univ. Rep., MD-12: 157 pp. Sutherland, A.J., 1967. Proposed mechanism for sediment entrainment by turbulent flows. J. Geophys. Res., 72: 6183--6194. Teleki, P.G., 1972. Wave boundary layers and their relation to sediment transport. In: D.J.P. Swift, D.B. Duane and O.H. Pilkey (Editors), Shelf Sediment Transport, Process and Pattern. Dowden, Hutchinson Ross, Stroudsburg, Pa., pp. 21--59. Thorn, M.F.C., 1974. Hysteresis of fine sand suspensions in a tidal estuary. Hydraulics Res. Sta. Note No. 17:3 pp. Vanoni, V.A., 1964. Measurements of critical shear stress for entraining fine sediments in a boundary layer. Calif. Inst. Technol. Rep. KH-R, 7. Wallace, J.M., Eckelmann, H. and Brodkey, R.S., 1972. The wall region in turbulent shear flow. J. Fluid Mech., 54: 39--48. White, C.M., 1940. The equilibrium of grains on the bed of a stream. Proc. R. Soc. Lond., 174: 322--338. Willmarth, W.W. and Lu, S.S., 1972. The structure of Reynolds stress near the wall. J. Fluid Mech., 55: 65--92;