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PRELIMINARY STUDIES ON MIXING AND ENTRAINMENT OF FLUID MUD
12. PERSONAL AUTHORS)
Mehta A.J.; Scarlatos, P.D.; Srinivas, R. 13a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Year. Montt,Day) 15. PAGE COUNT
Final FROM 2/28/89 TO7L3129 February, 1991 24 16. SUPPLEMENTARY NOTATION
17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)
FIELD GROUP SUB-GROUP Fluid mud
19. ABSTRACT (Continue on reverse if necessary and identify by block number)
PART I. A matter of particular interest to estuarine sedimentation problems is the dynamic behavior of fluid muds. It is suggested that shear flow-induced interfacial instabilities
leading to the erosion of fluid muds is a significant resuspension mechanism. Thus, the erosion process is controlled by the scale of near-bottom turbulence and the vertical velocity and density profile structures. Under strongly entraining conditions in which settling is small, the density profile microstructure of the entraining material is governed by buoyancy stabilized turbulent diffusion. A step-wise profile of the suspended sediment
is theoretically developed under simplified physical conditions. Field and experimental data support the theoretical assumptions, while experience from other stratified flow phenomena
provides valuable insight relative to studies required to quantify fluid mud erosion response to hydrodynamic forcing.
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19. ABSTRACT (Continued)
PART II. An experiment is described to simulate fluid mud entrainment by current shear. A
recirculating flume with a disk pump system to produce shear was used for this purpose. Significant interfacial instabilities were observed with resultant fluid mud entrainment. An empirical relationship is obtained between the non-dimensional buoyancy flux and the Richardson number, which is then compared with that obtained for salt-stratified flows. This comparison makes apparent the additional dissipation of turbulent kinetic energy in sedimentwater systems relative to salt-stratified systems, particular at higher Richardson numbers as the entrainment rate decreases substantially.
PRELIMINARY STUDIES ON MIXING AND ENTRAINMENT OF FLUID MUD
A. J. Mehta P. D. Scarlatos and
U.S. Army Engineer Waterways Experiment Station 3909 Halls Ferry Road Vicksburg, MS 39180-6199
ON MIXING AND ENTRAINMENT OF FLUID MUD
A. J. Mehta P. D. Scarlatos R. Srinivas Coastal and Oceanographic Engineering Department
University of Florida
336 Weil Hall
Gainesville, FL 32611
U.S. Army Engineer
Waterways Experiment Station
3909 Halls Ferry Road
Vicksburg, MS 39180-6199
Material presented herein constitutes the final report for the contract
DACW39-89-K-0012: Separation (Stratification) and Mixing of Fine-Grained Dredge Sediment Slurries A "Bench" Study, through the U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. The report is divided into two parts as follows. Part I: Some Observations on Erosion and Entrainment of Estuarine Fluid Muds, by. P.D. Scarlatos and A.J. Mehta (1990), has been published in Residual Currents and Long-term Transport, R.T. Chen ed., Coastal and Estuarine Studies Vol. 38, Springer-Verlag, New York: 312-332. Part II: Observations on Estuarine Fluid Mud Entrainment, by R. Srinivas and A.J. Mehta (1989) has been published in International Journal of Sediment Research, 5(1): 15-22.
TABLE OF CONTENTS
PREFACE ii LIST OF FIGURES iv PART I. Some Observations on Erosion and Entrainment of Estuarine Muds I
Abstract 1 Introduction 1 Physical Properties of Fluid Muds 2 Erosion and Entrainment 3
Interfacial response 3 Entrainment 5 Experimental observation 6
Microstructure and Density Profile 6 Concluding Remarks 9 References 10 PART II. Observations of Estuarine Fluid Mud Entrainment 13
Abstract 13 Introduction 13 Background 14
Interfacial response 14
Entrainment 14 Experimental Methodology 15 Experimental Observations 17 Conclusions 19 Acknowledgement 20 References 20
LIST OF FIGURES
FIGURE PAGE PART I.
1. Schematic definition of fluid mud relative to mud density
2. Characteristics of density and velocity profiles in the
interfacial region 4
3. High concentration mud underflow over Yellow River delta
(adapted from Wright et al., 1988). 4
4. Interfacial disturbance at RiU = 5 (dp/p1 = 0.007). Wave
height ranged from 1.3 to 2.5 cm, period from 3 to 5 s.
Grid squares are 2.54 cm2. 6
5. Microstructure of three sequential sediment vertical
profiles in the Severn estuary, England, taken over a
16 min interval (adapted from Kirby, 1986). 7 6. Simulated development of suspension density microstructure. 9
1. Definition sketch for density and velocity profiles in mud
stratified two-layer system. 15 2. Experimental flume. 16 3. Variation of mixed layer velocity profile with time. 16 4. Massive interfacial convolution (Ri, 7). 18 5. Non-dimensional buoyancy flux. Q, Vs Richardson number, Ri 19
Some Observations on Erosion and Entrainment of Estuarine Fluid Muds
Panagiotis D. Scarlatos1 and Ashish J. Mehta2
ISouth Florida Water Management District P.O. Box 24680
West Palm Beach, FL 33416-4680.
2Coastal and Oceanographic Engineering University of Florida
Gainesville, FL 32611
A matter of particular interest to estuarine sedimentation problems is the dynamic behavior of fluid muds. It is suggested that shear flow-induced interfacial instabilities leading to the erosion of fluid muds is a significant resuspension mechanism. Thus, the erosion process is controlled by the scale of near-bottom turbulence and the vertical velocity and density profile structures.
Under strongly entraining conditions in which settling is small, the density profile microstructure of the entraining material is governed by buoyancy stabilized turbulent diffusion. A step-wise profile of the suspended sediment is theoretically developed under simplified physical conditions. Field and experimental data support the theoretical assumptions, while experience from other stratified flow phenomena provides valuable insight relative to studies required to quantify fluid mud erosion response to hydrodynamic
Simulation of estuarine residual cohesive sediment transport leading to sedimentation is an important but also a complicated task. This complexity is largely defined by our level of understanding of a wide variety of near-bed flowsediment interactions which characterize the overall transport problem. Due to the small particle size, cohesive sediment particle movement is controlled not only by inertial, gravitational, buoyancy and drag forces, but also by electro-chemical ones. Since however a microscopic description of the processes involved in cohesive sediment transport is not presently feasible, for simulation purposes a macroscopic approach using lumped parameters is usually applied (Scarlatos and Partheniades, 1986).
During a tidal cycle or an episodic event, spatial and temporal redistribution of the suspended sediment concentration occurs by virtue of the effects of the different phases of the sedimentation cycle. From field and laboratory observations it has been noticed that, generally, as shown in Figure 1 the vertical concentration (density) profile of fine sediments can be categorized into three zones: an upper zone of mobile, relatively low concentration suspension, a middle zone of higher concentration suspension or fluid mud, and the lower zone of settled bed (Kirby and Parker, 1983). The fluid mud zone is of particular practical importance because, due to its high concentration and weak internal structure, it can be easily entrained and can substantially contribute to turbidity even under relatively low energy inputs (Ross, 1988). Therefore, assuming an initially sharp interface between the overlying water and fluid mud during quiescent periods as occurs for example during slack water, it is important to understand the erosion behavior of the mud-water interface, and the evolution of the interface as the fluid mud entrains into the water column due to shear flow.
Coasetaand Estuarine Studies, Vo 38
R.T CWVng (Ed.)
lesdual Currents and Long-term anspot
a Spinger-Verlag New Wrk, nc., 1990
Many past studies on cohesive sediment transport seem to have neglected the fluid mud zone, emphasizing instead the erosion of the deposited bed (Mehta, 1986). Such has been the case in spite of the fact that it has been long recognized that fluidized muds do not erode in the same manner as a settled bed (Parker and Kirby, 1982). Yet, because of inadequate knowledge of the underlying mechanism for fluid mud erosion, numerical models have had to rely on conjecture and empiricism (Ariathurai, 1974; Hayter, 1983). The purpose of this paper is to briefly describe the physical properties of fluid muds, to review erosion and entrainment processes of initially stationary high concentration suspensions, to present a simple model for simulation of vertical suspension density distribution which results from entrainment, and, thereby, to highlight some areas for further research.
II. Physical Properties of Fluid Muds
The suspended sediment concentration typically increases rather dramatically downwards from the surface to the bottom of the water column (Figure 1). The concentration profile may be smooth or step-wise depending on, among other factors, the concentration of total sediment mass in suspension (Kirby, 1986). The transition from a mobile suspension of low concentration (< 300-500 mg/1), to a stationary suspension of high concentration (> 10 g/1), usually occurs within a very narrow vertical distance, creating thus a steep density gradient. The zone of the steepest gradient within the water column is called the lutocline layer (Figure 1). The thickness of the various zones and the degree of vertical stratification depend on the magnitude of energy input by the tides, wave action, wind stress, geophysical motions, bioturbation or sometimes even human activity.
FIGURE 1. Schematic definition of fluid mud relative to mud
Fluid muds are usually contained between the lutocline and the settled bed. The thickness of the fluid mud layer may be. from a few centimeters to. several meters. Typical concentrations for fluid muds are usually within the range of 10 to 320 g/l, which- correspond to bulk densities approximately between 1.03 to 1.2 g/cm3, respectively (Bradley et al., 1988).. However, as Been and Sills (1981) have noted from their laboratory work on- quiescent settling, a weak solid structure is usually present when the concentration exceeds about 220 g/l.
Fluid muds exhibit a strong time-dependent, non-Newtonian fluid response (Bryant et al., 1980); therefore any realistic description of their dynamic behavior must account for this essential property. Various constitutive models have been used for simulation of their mechanics including kinematics and energy dissipation. These include: elastic, poro-plastic, viscous fluid, Bingham plastic, and viscoelastic models (Maa, 1986). Laboratory and field tests indicate that the rheological
behavior of fluid muds, i.e., the relationship between stress and rate of strain, may change from pseudoplastic at relatively low concentrations, to Bingham plastic at high concentrations (Sills and Elder, 1986; Maa, 1986). The Bingham yield stress
documented in high concentrations provides an indication of the work required to disrupt the floc structure in the cohesive .suspension. Under very high shear stresses fluid muds approach Newtonian behavior, i.e., a linear relation between stress and rate of strain. This is due to the fact that increasing shear rates
break down the flocs and reduce the particle's partial support by the cohesive structure.
Besides concentration, the rheological properties of fluid muds are greatly influenced by the physico-chemical characteristics of the supporting fluid and the composition of the sediment, which essentially indicates the importance of cohesion in defining mud behavior. Experimentally it has been observed that, for example, very weakly cohesive kaolinite-quartz mixtures tend to behave like a pseudoplastic material, while highly cohesive estuarine muds may apparently behave as Bingham plastics (Krone, 1963). Very few muds however seem to be truly Bingham plastics; most exhibit pseudoplastic character at very low rates of shear, and consequently creep under even mild forcing due to shear flow or gravity (Bryant et al., 1980; Kirby, 1986).
III. Erosion and Entraine:
III.1 Interfacial response
Under low energy conditions, such as during slack water, when the suspended sediment settles rapidly, fluid mud is formed near the bed, provided the rate of deposition is high enough to prevent dewatering of the underlying material. The thickness of the fluid mud-water interface depends on the concentration, the particle size distribution, and the time from the initiation of deposition. Based on these physical characteristics, Davis and Hassen (1988) estimated the spreading of the interface during deposition by considering hindered settling effect, particle interaction, and a slight polydispersity of the suspension. Hindered settling effect, caused by increasing sediment concentrations in the lower layers and very
important in influencing sediment dynamics near the bed, was shown to lead to a decrease of the interface thickness.
Given a two-layer flow with a distinct interface, erosion of the fluid mud layer
is believed to be controlled by the mechanisms of shear flow interfacial instability. This is unlike, for example, the erosion of a settled bed which erodes by dislodgement and entrainment of surface flocs due to applied stress. Liu (1957) was first to suggest the notion of interface instability in sediment transport studies, while attempting to explain ripple formation in moveable beds.
Unfortunately his application may not have been entirely appropriate, because the motion of granular beds is more discrete than continuous, as identified by rolling and saltation of individual grains. On the other hand, fluid mud and the overlying water column can be treated in many respects as a system of two miscible continua with different densities and different rheological properties separated by an interface of a certain thickness.
In general, under shear flow conditions, the thickness of the density interface
(d) affects the mode of instability that will erode the interface. The mode of
instability is equally affected by the thickness (D) of the region of high velocity gradient (Figure 2). In case the mid-axis of the velocity and density gradients coincide, then when d is approximately equal to D, the prime mode of instability is that of the Kelvin-Helmholtz type; when d is smaller than D, the instability- is of
the Holmboe type. The main characteristic of the Kelvin-Helmholtz instability is the roll-up and pairing of the interfacial vortices (Delisi and Corcos, 1973). The
Holmboe mode of instability is recognized by the sharp crests along the which finger alternatively into both fluids (Browand and Wang, 1972).
VELOCITY OR DENSITY
FIGURE 2. Characteristics of density and velocity interfacial region.
profiles in the
Large scale internal wave instabilities over high concentration mud underflows have recently been recorded in the Bohai Gulf offshore near the mouth of the Yellow River, China (Wright et al., 1988). These features (Figure 3), thought to be ubiquitous, were of rather high frequency (2.5-5.0 x 10 3 Hz) close to the local Brunt-Vaisala frequency (3.3 0.8 x 10 3 Hz). They were generated by shear arising from the relative motion between the gravity underflow and the overlying water, and were believed to contribute to underflow deceleration. Undoubtedly, such large disturbances are likely to play a significant role in mixing the mud flow with the upper fluid in which another, more diffused, low concentration plume was observed.
1 2 3 4 5 6 7 8 9 10
DISTANCE FROM MOUTH (km)
FIGURE 3. High concentration mud underflow over Yellow River
delta (adapted from Wright et al., 1988).
Instability can be studied by superimposing small perturbations on the mean flow, and representing the continuity and Navier-Stokes momentum equations in terms of the perturbated quantities. By using the theory of normal modes, the linearized governing equation for stability is given by the well-known Orr-Sommerfeld equation (Betchov and Criminale, 1967). The flow is considered as stable if the perturbations decay in time. Analytical solutions of the Orr-Sommerfeld equation are feasible only when the two-phased fluid system under consideration is either inviscid-inviscid or inviscid-viscid, i.e., when there is a velocity .discontinuity at the interface. The case of interfacial stability between viscous-viscous fluids requires numerical treatment of the governing stability equation. The case of instability, where the velocity and density gradient axes do not coincide, have been studied by Lawrence et al. (1987).
0 .0/.mu Internal Waves Delta-Front Ri- I I Ii I I I
The analytical results obtained from the Orr-Sommerfeld equation provide an understanding of the conditions that lead to instability and breaking of the interface. These results are given in terms of the physical flow characteristics such as flow velocity, viscosity, surface tension, and density variations. in the system. For inviscid-viscid flow, which can be considered in a very simplified treatment of the water-fluid mud type problem, the linear stability criterion is given as
(2T3/2/[V2P2(gAP)1/21)]/2 1 (1)
where T is the surface tension, v is the kinematic viscosity of the upper fluid, g is the acceleration of gravity, Ap P2-P1, and pl, P2 are the densities of the upper and lower fluid layers, respectively. Eq.(l) was generated considering the fact that the time scale of the perturbation must be much shorter than vorticity diffusion in the mean flow. It is illustrative of the number of characteristic parameters which are involved even in this simplified approach. The nature of the assumption made concerning the physical properties of the two fluids critically influences stability. Thus, for example, the stability criterion for viscidinviscid fluid systems is more restrictive than the criterion for inviscid-inviscid fluids (Hogan and Ayyaswamy, 1985).
A realistic assessment of the instability of the water-fluid mud interface, presently unavailable, requires a more rigorous approach which considers not only the viscous effects, but also the rheological properties of the fluid mud. The nonNewtonian nature of fluid mud complicates the instability problem by introducing variable viscosity, non-zero normal stress differences, and out of phase relationships between stress and rate of strain (Maa, 1986). Depending on the magnitude of the characteristic flow velocity, characteristic length and fluid relaxation time, the fluid mud system can be simplified and treated either as a nonNewtonian fluid with non-zero normal stresses and in-phase stress versus strain rate, or with zero normal stresses and out-of-phase stress versus strain relationship (Harris, 1977). Attention must also be given to the fact that Squire's theorem, which states that two-dimensional flows provide more strict instability conditions than three dimensional flows, may also not be valid for non-Newtonian fluid and, consequently, mud flow. These issues are the subject of ongoing 'research at the University of Florida, and will be examined in a subsequent publication.
Once the interface breaks, the sediment particles entrain into the overlying water column, and there is mixing between the fluid mud and the overlying water. Entrainment processes have been extensively studied for the cases of thermal and salinity stratification, but apparently not for fluid muds. However, the role of this mechanism in modeling fine sediment resuspension has been recognized by, among others, McLean (1985). Entrainment is usually quantified as a function of the bulk Richardson number, Ri, which expresses the balance between buoyancy and inertia forces. There are different laws pertaining to the rate of entrainment based on the degree of stratification. The entrainment velocity, ue, can for example be normalized either by the mean flow velocity, U, the shear velocity, u*, or the velocity jump, dU. Accordingly, depending on which of these is used as the characteristic velocity, the Richardson number is defined as RiU Ri*, or RidU. Narimousa et al. (1986) suggested the shear velocity as the scaling velocity, and proposed the expression:
ue/u* aRi*b (2)
where a and b are numerical coefficients defined accordingly as: a 0.65, b -1/2 for 15< Ri <150, a -7.00, b -1 for 150< Ri, <800, and a 5.00, b -3/2 for 800< Ri*. Tiese exponential laws can be explained on physical grounds as follows: for low Ri, shear is high and the density gradient is small, so that mixing is produced by the larger, energy carrying eddies. For intermediate values of Ri-, the processes are less chaotic, so that entrainment is more ordered and caused by Kelvin-Helmholtz instability. Finally, for large Ri shear is very weak and entrainment is due primarily to the kinetic energy ol the turbulent layer. If the density gradient is extremely large, then entrainment is supressed and the only mixing is caused by molecular processes. In another publication, Narimousa and
Fernando (1987) indicated that' the best scaling velocity is the velocity jump at the interface. The explanation of their suggestion was based on the fact that the energy for turbulent mixing of the interface is produced at this high shear zone.
III.3 Experimental observation
Five very preliminary experimental runs were performed at the University of Florida in order to monitor the interfacial instability of a two-layered flow of clear water over kaolinite-water suspensions. The tests were conducted in a 18.3 m long and 0.6 m wide recirculating flume. A description of this flume is given by Dixit (1982). Initially the kaolinite-water suspension was thoroughly mixed through high speed velocity, and then by reducing the velocity substantially, the sediment was allowed to settle. After 10-20 minutes when a distinct interface was formed, the velocity was increased to a predetermined magnitude and the interfacial activity was filmed for a duration of approximately 30 seconds. The initial density jump, dp/p1 (P2-P1)/P1, was varied from 0.005 to 0.009.
Results from these experiments showed that for RiT 19.0 there was a distinct wave at the interface but no breaking (no turbulenE entrainment). For RiU 5.0 (Figure 4) there was breaking of the wavy interface, while for R 1.7 the wave breaking caused smoke-like wisps that were entraining into eupper layer, and resembled the Holmboe type of instability. Furthermore, the interfacial thickness, d, was found to be on the order of 5% of the depth of the mixed upper layer. The observed value of RiU 19.0 for stability (non-entrainment) of a water-mud system coincides with the experimental data presented by Narimousa 'et al. (1986), for fresh water-salt water system. However, more data are necessary in order to establish confidence in the range of applicability of the existing state-of-the-art
understanding of instability and entrainment to the case of fluid mud systems, as suggested by Odd and Cooper (1989).
Qualitative differences (e.g., in the meaning of coefficients a and b in Eq.(2)) are likely to arise in replacing salt water by fluid mud due to differences in the rate of dissipation of turbulent kinetic energy in these two fluids, as suggested by the recent laboratory results of Srinivas (1989) and Wolanski .et al. (1989), in which sediment-induced collapse of turbulence was observed at the water-fluid mud interface.
Wave Brek In.
FIGURE 4. Interfacial disturbance at Riu 5 (dp/p1 0.007).
Wave height ranged from 1.3 to 2.5 cm, period from 3 to 5 s. Grid
squares are 2.54 cm2.
IV. Hicrostructure and Density Profile
Continuous monitoring of the vertical density profile in stratified estuaries has indicated the occurrence of a step-wise profile structure. This microstructure has been identified in cases of salinity, temperature, and sediment stratification. Typical profiles of suspended cohesive sediment, recorded a few minutes apart in the Severn Estuary in England, are shown in Figure 5 (Kirby, 1986). The time scale of
the microstructure development, in the differentiated (stratified) layer between the mixed upper layer and the main or primary lutocline near the bed, was on the order of 10 to 15 minutes. The general location of the primary lutocline is largely controlled by hindering effects during settling (Ross, 1988), while turbulent diffusion strongly influences mixed layer thickness. Posmentier (1977) presented data for salinity microstructure in the Hudson River with a longer representative time scale on the order of 90 minutes. Another difference between Kirby's and Posmentier's data is that the -former documented a dynamic system where the number and position of secondary lutocline steps changed in time, while the latter presented a profile where the microstructure apparently stabilized after a certain time. Since the microstructure is indicative of the vertical stratification process, its understanding is essential for quantifying sediment mass exchange between stationary and mobile sediment suspensions.
1.- 10- Secondary CL Lutocline
I I I fill
0 -5 9
CONCENTRATION (gI 'I)
FIGURE 5. Microstructure of three sequential sediment vertical profiles in the Severn estuary, England, taken over a 16 min
interval (adapted from Kirby, 1986).
A full understanding of the mechanisms of density microstructure have not as yet been established. There are a number of theories which attribute the step-wise density structure to different causes such as Kelvin-Helmholtz instability, breaking of internal waves; viscous overturning, or salt fingering. A common agreement among these theories is that the microstructure is due to turbulent energy patches resulting in vigorous and uniform mixing which creates sharp gradients with the rest of the density layers. Since the representative length scale of the phenomenon is the profile step, the turbulent eddies affecting the system are expected to be of the same magnitude (Monim and Ozmidov, 1986).
Theoretical analysis of the vertical suspension density distribution under general conditions is quite complicated. Assuming steady state flow conditions, homogeneity along the longitudinal axis, and considering only the entraining phase .of flow when gravitational settling effects may be ignored for simplicity, the vertical concentration profile can be studied very approximately by means of the one-dimensional diffusion equation written as
.C -F (3)
where C is the sediment concentration, F is the sediment flux, z is the vertical axis, and t is time. A comma in front.of a subscript denotes partial differentiation with respect to that subscript. Applying Fick's law for turbulence induced fluctuations the sediment -flux reads
where K is the coefficient of vertical diffusivity for mass. In sediment stratified systems turbulence is strongly affected by the stratification characteristics (McLean, 1985). Therefore, the buoyancy stabilized coefficient of diffusivity can be expressed by a semi-empirical form as
K A (1 + sRi )r (5)
where A is the coefficient of vertical eddy diffusivity under neutral stability, and s, r are empirical coefficients (Okubo, 1970). The coefficient A is related to the velocity and density fluctuations in the vertical direction. Using the mixinglength theory, A can be approximated as
A = (1.)2 ju J (6)
where 1. is the mixing length in a neutrally stratified flow, and u is the depth average horizontal velocity. For fluid muds the Richardson number can be expressed as (Scarlatos and Mehta, 1988)
Ri (g C [(ps p )I/p )/u2 (7) ,z s w w ,z
where p p are respectively the density of water and sediment. Assuming for simplicity a logarithmic profile for the velocity, the mixing length is given as
1. (kz [1 (z/H)])1'2 (8)
where k is the von Karman's constant, and H is the depth of the boundary layer. Setting X C, Eqs.(3-4) yield
C,t -F C (9)
Due to the existence of the gradient F Eq.(9) is a nonlinear partial differential equation. Posmentier (1977) suggested'Ehat when the factor F, is negative then the system becomes unstable. As a result of the instability, there is thorough mixing which creates steep density gradients and a more stable system. When stability increases the factor F, becomes positive. Combining Eqs.(4-8) yields
F R [I + (I + r)BX] (1 + BX)r-1 (10)
where R and B are defined as
R kz [1 (z/H)] ju J (11)
B s g[(p5 )/p]/U2 (12) s w w ,z
Since R is always positive, the sign of the quantity F, depends on the sign of the remaining right hand side of Eq.(10). From field observations it has been estimated that the values for the coefficients s and r are approximately between 3.33 < s < 10 and 2.0 < r < 0.5, (Ross, 1988).
It is not known whether all the values between the upper and lower inequality limits of s and r are physically realistic. For r -1, F, is always positive, and therefore Eq.(9) is well-posed. In. this case, it is not possible to generate instability and consequent microstructure. For r -2 the stability requirement is given as
(BX)2 (sRi U)2 < 1 (13)
Assuming s 4, Eq.(4) yields Ri < 1/4, which is the well-known stability criterion for shear flows (Okubo, 1970). E similar expression can be derived for r -0.5. Whenever there is violation of the stability criterion, internal waves break and a patch of weak turbulence is generated. This turbulent patch spreads horizontally and creates a layer of homogeneous water which causes the generation of microstructure.
F - KC
Simulation of the microstructure development process can be achieved by solving Eqs.(9-12). The numerical scheme used for the solution was the Newton linearization iterative technique which applies to the governing equation in a form of:
C G(z; t; C; C ; C ) (14)
zz z ,t
Then, Eq.(14) is expanded as
Cn+l)- G(n) + C(n+l)- C ] G,( + [Cn+) C
zz [I ,C z z ,X
+ [C(n+l) ,] G (15)
where n is the number of iterations (Ames, 1977). The term C is discretized by using a central difference scheme. For this numerical scheme'Re solution matrix is tri-diagonal while the iteration converges quadratically. Boundary conditions for the system must be specified at the free water surface and at the consolidated bed. At the free surface the concentration is assumed as constant while at the bed a mass flux is defined to essentially account for net erosion of sediment (Mehta, 1986).
Simulated result for a hypothetical case in a 20 m deep unit width channel is presented in Figure 6. In this simulation a logarithmic velocity profile with a surface current of 0.2 m/s was used, and the values for the coefficients s and r were s 4.0 and r -2.0. These values for s and r are typical values estimated from field data (Ross, 1988). The time step used was 5 s while the flow depth was discretized into 80 segments for one run and into 50 segments for another. The microstructure generation was however not measurably affected by the number of segments over the depth. It can be clearly seen that starting from a smooth profile, a step-wise profile developed after 90 s of simulation time. The lutocline steps defining the differentiated layer are not however as dramatic as those recorded in nature (Figure 5). Neglect of horizontal advection may be one reason for this discrepancy. For any velocity profile other than logarithmic, e.g. for a bottom reverse current, the vertical mixing coefficient must be redefined since Eqs.(4-8) will not be valid. In general, the mechanics of generation of the differentiated layer is the same as that in a halocline, and is physically explained by the non-linear dependence of diffusion flux on the density gradient (Postmentier, 1977; Ross, 1988).
z1H t=0 t=90s
FIGURE 6. Simulated development of suspension density
V. Concluding Remarks
It appears that fluid mud erosion and entrainment mechanisms can be simulated by approaches applied to salinity or thermal density flows. Erosion of fluidized mudwater interface depends strongly on the velocity and density gradients, and resembles interfacial instability phenomena typical to stratified flows
characterized by haloclines and thermoclines. Laboratory experiments with water and kaolinite-water suspension seem to show a qualitative agreement with other existing instability data. However, due to differences in the rate of turbulence dissipation
near the interface, entrainment rates of fluid muds, while undoubtedly Richardson number dependent, are likely to show qualitative differences from those for salt and heat.
The vertical profile of the mobile suspension at the upper water levels exhibits a step-like density microstructure. This microstructure can be simulated very approximately by using a diffusion type equation with an appropriate value for the vertical mixing coefficient. The simulated density "steps" are not however as dramatic as those observed in nature. The discrepancy may result from the neglect of horizontal advection. In addition, the use of the Fickian closure approach is inherently limited by empiricism, and a higher order closure scheme may be essential for quantitative simulation.
Due to the non-Newtonian nature of fluid muds, additional theoretical and experimental research is needed on the instability and entrainment mechanisms in shear flows arising from hydrodynamic forcing under generalized conditions of outof-phase stress versus strain rates, variable viscosity, and non-zero normal stress differences.
Support from the U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS through contract DACW39-89-K-0012 is acknowledged.
Ames, W. F., 1977: Numerical Methods for Partial Differential Equations. Academic
Ariathurai, R., 1974: A finite element model for sediment transport in estuaries.
Ph.D. Thesis, University of California, Davis.
Been, K., and G. C. Sills, 1981: Self-weight consolidation of soft soils: an
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Bryant, R., A. E. James and D. J. A. Williams, 1980: Rheology of cohesive
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Delisi, D., and a. M. Corcos, 1973: A study of internal waves in a wind tunnel.
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dispersal and deposition of Nature, 332(6164), 629-632.
rnternatiOnal Journal of SOeinment Research
Volume 5, No. 1. January 1990
OBSERVATIONS ON ESTUARINE FLUID MUD ENTRAINMENT
Srinivas, R. 1and Mehta, A. J2.
An experiment is described to simulate fluid mud entrainment by current shear. A recirculating flume
with a disk pump system to produce shear was used for this purpose. Significant interfacial instabilities were observed with resultant fluid mud entrainment. An empirical relationship is obtained between the non-dimensional buoyancy flux and the Richardson number, which is then compared with that obtained for salt-stratified flows. This comparison makes apparent the additional dissipation of turbulent kinetic energy in sediment-water systems relative to salt-stratified systems, particularly at higher Richardson numbers as the entrainment rate
Key Words: Estuary, Mud entrainment, Fluidized mud, Instability, Interface, Richardson number,
Stratified flow, Disk pump, Buoyancy flux, Shear flow.
Prediction of mud bed erosion by forcing due to tidal currents usually requires a numerical solution of the advection-dispersion equation for sediment mass transport. Key role is of course played in this by the bottom boundary conditions defining erosion and deposition fluxes. The issue of erosion is briefly considered here. .noting that it is customary to calculate the rate of erosion as a function of the bed shear stress in excess of the erosion shear strength of the bed (Mehta et al., 1982).
While it is found from practice that such erosion rate expressions work reasonably well for numerical model application in low to moderate concentration environments (Hayter and Mehta, 1986), questions regarding their applicability persist in cases involving high concentration fluid muds (Odd and Cooper, 1989). Fluidized mud has effectively no shear strength, and the mud-water interface is easily destabilized under shear flow. It is therefore instructive to focus attention on what might be the entrainment behavior of the fluidized mud layer underlying initially sediment-free water which starts
1. Graduate Research Assistant, Coastal and Oceanographic Engineering Department, University of Florida,
Gainesville. FL 32611, U.S. A.
2. Professor. Coastal and Oceanographic Engineering Department, University of Florida, Gainesville, FL
from rest, thus shearing the mud-water interface.
Under low energy conditions, such as during slack water, when suspended sediment rapidly settles, fluid mud is formed near the bed, provided the rate of deposition is high enough to prevent rapid dewatering of the freshly settled material. Erosion of this mud layer and its entrainment into the overlying water column following slack water is essentially controlled by the mechanisms of shear flow-induced interfacial instability. In general the thickness of the density interface, d affects the mode of instability that will erode the interface. The mode of instability is equally affected by the thickness, D, of the region of high velocity gradient (Fig. 1). In case the mid-axis of the velocity and density gradients coincide then, When d is approximately equal to D the prime mode of instability is that of the Kelvin-Helmholtz type; when d is smaller than D, the instability is of the Holmboe type. The main characteristic of the Kelvin-Helmholtz instability is the roll-up and pairing of the interfacial vortices (Delisi and Corcos, 1973). The Holmboe mode is recognized by the sharp crests along the interface which finger alternatively into both fluids (Browand and Wang, 1972).
Once the interface breaks, the sediment aggregates entrain in the water column, and there is mixing between the fluid mud and the overlying water. Entrainment processes under unidirectional shear flows have been extensively studied for the cases of thermal and salinity stratification, but apparently not for fluid muds. However, the role of this mechanism in modeling fine sediment resuspension has been recognized by, among others, Wolanski and Brush (1975), McLean (1985) an Odd and Cooper (1989). Entrainment is usually quantified as a function of the Richardson number, Ri, which expresses the balance between buoyancy and inertia forces. There are different laws pertaining to the rate of entrainment based on the degree of stratification, since the entrainment -velocity can be normalized either by the mean flow velocity, the shear velocity, or the velocity jump.
However, the velocity jump across the interface, AU, appears to be the most significant velocity scale for entrainment across sheared density interfaces, as shear production in the entrainment zone is mainly responsible for turbulent mixing. Turbulence is also produced at the sidewalls, but most of this is dissipated at the walls themselves, with the portion diffusing outwards being only a small fraction of the total amount (Hinze 1975), as confirmed by Jones and Mulhearn (1983). Visual observation of injected dye-lines seemed to indicate only minimal diffusion of momentum below the level of the density interface. This was as-well the case with Narimousa and Fernando (1987). It was thus decided to use the mean velocity of the turbulent mixed layer, u for scaling purposes. Thus, the Richardson number for the problem at hand was defined as Ri. = hAb/u2, where h is the depth of the mixed layer, Ab is the interfacial buoyancy step; buoyancy being defined as b = g (p1 po) /po, with pi and po being
Srinivas, R. and Mehta A. J.
the density of the fluid and a reference density, respectively.
Profile,p(z) Profile, u(z) Mixed Layer
Fig. 1 Definition sketch for density and velocity profiles in mud stratified two-layer system
IV. EXPERIMENTAL METHODOLOGY
Laboratory experiments were carried out in a plexiglass "race track" flume (Fig. 2), in order to investigate the influence of Ri. on entrainment of fluid mud (Srinivas, 1989). Similar flumes have been used successfully previously by other investigators to study salt stratified flows (Narimousa and Fernando, 1987). In this apparatus the initially clear water layer can be made to flow relative to the higher density fluid below by means of disk pump system, which is designed to minimize any intrusive effect of pumping on the interfacial dynamics.
The annular geometry at the flume ends is meant to gradually turn the flow without undue interference. As the width of the flume is necessarily large at the pump section, triangular plexiglass flow separators were placed upstream and downstream of the pump-section so as to create two equal channels of half the width elsewhere. The 200 cm long straight test section was used for observations.
The disk pump consists of two vertical shafts driven by a motor and rotate in opposite directions. Each shaft is stacked with a number of thin disks of sand blasted plexiglass. These are of two diameters, 4 and 13 cm, stacked alternately on each shaft, and so arranged that a small disk on one shaft meshes with the larger of the other.. thus almost sealing the center of the pump, while the fluid is thrown in a series of horizontal jets around the outside of the sfnaller disks and between the larger disks. In order to prevent the disks from "sucking up" the density interface, horizontal partitioning of the flume became essential in the region in proximity of the pumps. This "splitter plate" was extended
Observations on estuarine fluid mud entrainment
3 s5 F ow I Flow Seperator 148 4D Seperator
4 0 5
0.32cm Plaxigless 1.27cm Piexigiass
Fig. 2 Experimental flume
VELOCITY IN MIXED LAYER, u (cms 1)
FIg.'3 Variation of mixed layer velocity profile with time
into the downstream curved flume segment in order to minimize the effect of helical secondary flows on the processes occurring within the linear test section. An impeller had to be added just upstream of
mi;. cms-1 =2 U =6.6 t2 =8 U2 =9.9 13 =11 a3 =9.6 14 =18 4 =8.6 Is = 28 G = 7.8 16 = 42 U6 =7.8
Srinivas. R. and Mehta A. J.
the test section and above the splitter plate to enhance disk-generated velocities.
The investigation herein reported was restricted to kaolinite and bentonite as sediments. The initial bulk density of fluid mud was varied over 1. 03 -1. 08 gcm-3. The lower limit was selected such that the mud would be in the hindered gravitational settling range, which is critical for mud' s existence in the fluidized state while the upper limit was dictated by the performance constraints of the experimental set-up.
The flume was first filled to the requisite pre-selected height of tap water. Pre-mixed fluid mud was then introduced through the intake at the flume bottom. The resulting fluid mud layer underneath water was nearly homogeneous initially. The initial height of the fluid mud layer was generally always kept at just under the elevation of the splitter plate.
As the pump was started to the required rotation rate, it took -2. 5-3. 0 min for the inertia of the system to be overcome as the upper layer mean velocity increased to its peak value. With time, as the fluid mud was entrained, the density interface moved downwards. The cumulative effect of this increase in mixed layer depth and increasing dissipation of turbulent kinetic energy on the entrained sediment aggregates was a slow decrease in the mixed layer mean velocity. Fig. 3 is an example of the variation in the mixed layer velocity profile with time.
V. EXPERIMENTAL OBSERVATIONS
The experiments were carried out over a range of Ri. from about 4 to 30. Higher velocity values required for lower Ri. values could not be achieved. When the pump was turned on, the energy of the generated turbulent shear flow resulted in the development of substantial instabilities at the density interface. At low Ri. (<5), entrainment appeared to be turbulence dominated, and the base of the mixed layer appeared to be turbulent as well. This resulted in a rather diffuse and highly irregular interface. As Ri increased to above ~ 5, the interface became better defined and convoluted with massive, irregular undulation; see Fig. 4. Entrainment appeared to be dominated by wave breaking in which wisps of fluid were ejected into the turbulent mixed layer. Interspersed with these breaking waves were also some large-amplitude solitary-type waves which decayed without breaking (see also Narimousa and Fernando, 1987). The frequency and amplitude of these disturbances decreased with increasing Ri. and the interface became more regular. When the Ri. was above 10, the disturbances could be seen to grow slightly in amplitude, sharpen into non-linear crests and disappear suddenly as the "roller-action" of an eddy sheared off the crest (similar to Moore and Long, 1971). Beyond Ri. of 25, the intensity of entrainment was highly diminished and appeared to taper off.
Thus, the effect of turbulence in the upper layer was entrainment -of the stably stratified fluid mud across the density interface, with resulting buoyancy transfer. It can be shown (Moore and Long, 1971) that the buoyancy flux is essentially the same as the rate of change of potential energy per unit mass (of the entire system). Using a result form Kato and Phillips (1969), it can further be shown that the non-dimensional buoyancy flux is equivalent to the non-dimensional rate of deepening of the mixed layer due to entrainment alone, which is well understood to be a function of the relevant
Observations on estuarine fluid mnud entrainment
20 Srmnivas. R. and Mehta A. J.
Fig. 4 Massive interfacial convolution ( Ri. -7)
Richardson number. The buoyancy flux is q = dm/dt g/po, where dm/dt is the mass flux into the turbulent mixed layer, and po is the mean mixed layer density. Thus, we define the non-dimensional buoyancy flux to be Q = q/ (uAb) : see Srinivas (1989).A plot of Q as a function of Ri. is presented in Fig. 5. Notwithstanding obvious data scatter, a trend line can be drawn through the data according to the relationship Q = A Ri.-0-/[B2 Ri.21,
where A 0. 27, B = 20 and m = 0. 66. It is believed that the main effect of the splitter plate on this relation might have been to cause some discrepency in the values of the constants (A, B and m) of the above relation, and data scatter particularly at low Ri., but that the.basic relationship between Q and Ri. seems to be adequately quantified. The dashed line relation indicates a trend Q oc Ri.- (2) with n =0. 9, which is similar to the relation obtained by experimenters with salt-stratified systems with n =1 (e. g., Moore and Long, 1971: Narimousa and Fernando, 1987).
The n = 1 relationship can be considered in terms of energy changes. Kato and Phillips (1969) showed that the rate of increase of potential energy per unit mass is related to the rate of dissipation of turbulent energy per unit mass. It is possible to obtain from their equations the relationship dvj/dt ~= ul, /h (3) where V I is the potential energy per unit mass and u. is the friction velocity. Batchelor (1953) found that the rate of change of turbulent kinetic energy per unit vloume in isotropic turbulence, e, is given by
e = d(u) /dt~ () 3/ (4) where 11 is the length scale of the energy containing eddies, and U the rms turbulent velocity.
Assuming that u. =-5-u (Towmsend, 1956: Moore and Long, 1971), and that the homoge-
Srinivas, R. and Mehta A. J.
Observations on estuarine fluid mud entrainment
A =0.27 B=20.0 m=0.66
0 I0.9-' .f
O=ARI I('/ + Rt N Dentonite
- A R
S a 1 1 t oi
I I I I lI II
RICHARDSON NUMBER, RIu
Fig. 5 Non-dimensional buoyancy flux. Q, Vs Richardson number, Ri. neous layer depth h is of the the size of the energy containing eddies, e () 3/h
dV,/dt ~= (U) 3/h -.. e (6) From our experiments, Q = q/ (izb) = K/Ri. = K (U) IhAb, where K is a proportionality constant. Because d V, /dt = q therefore, d V, /dt= K() 3/h and thus d V, /dt- (u)3/h, as in the form of eq. 6. This evidence seems to indicate that in geophysical situations and similar experiments, q (~ dV1/dt) ~. e with the "constant" of proportionality probably depending on the coefficients of viscosity and diffusivity.
In the present experiments, at low mixed layer concentrations, there apparently was not much additional dissipation of kinetic energy to counteract the downward buoyancy flux due to sediment fall velocity, as indicated by the trend of eq. 2 with n = 0. 9 (as compared with the n =1 trend for saltstratified systems). However, at Richardson numbers greater than -25, the buoyancy flux is observed to fall off much more drastically when, it is surmised that, a greater fraction of the input energy is used up merely in maintaining the sediment particles in suspension in the mixed layer.
The hitherto neglected phenomenon of fluid mud entrainment due to unidirectional current shear in an estuarial environment prompted this study. It appears that the non-dimensional buoyancy flux,
i i i i i
j j - - - -
Q, is related to the relevant Richardson number, Ri., by the relation described by eq. 1 for the range of Richardson numbers considered. This relationship indicates that at low Richardson numbers, the equation is similar to that obtained for many salt-stratified systems. which can be further shown to indicate that the rate of increase of potential energy of the system is of the same order as dissipation. However, for Richardson numbers >-25, the trend line indicates a much steeper fall off in the dimensionless entrainment rate than for salt-stratified experiments, indicating additional dissipation of kinetic energy as the mixed-layer concentration increases.
This study was supported by the U. S. Army Engineer Waterways Experiment Station, Vicksburg, MS (contract DACW39 -89 -K -0012).
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