CHEMICAL TRANSPORT IN A SORBING WAVE BOUNDARY LAYER

Chiu-On Ng

Department of Mechanical Engineering

The University of Hong Kong, Pokfulam Road, Hong Kong

Tel: (852) 2859 2622; Fax: (852) 2858 5415; E-mail: cong@hkucc.hku.hk

Abstract: The transport of a chemical species under the pure action of surface progressive waves in the benthic boundary layer that is heavily loaded with sorbing particles is studied theoretically. The flow structure of the boundary layer is approximated by that of a two-layer Stokes boundary layer with a sharp interface between clear water and a heavy-fluid. For a thin layer of heavy-fluid, whose thickness is comparable to the surface wave amplitude and the Stokes boundary layer thickness, effective transport equations are deduced using an averaging technique based on the method of homogenization.

Keywords: Stokes boundary layer, mass transport, dispersion coefficient

1  INTRODUCTION

Transport of littoral sediment due to the combined action of waves and currents has been extensively studied (e.g., see Fredsfe and Deigaard, 1992; Nielsen, 1992; Madsen, 1993, and the references therein). Typically, wave action is responsible for keeping sediments in suspension, while currents provide the net transport. Theoretical models of sediment transport where waves play a direct role in both aspects of entrainment and net transport have been presented by Mei and coworkers (Mei and Chian, 1994; Mei et al., 1997 and 1998). In their dispersion models, the sediment concentration is assumed to be so low that the flow characteristics are essentially not affected by the particles. This assumption however breaks down when the solid concentration becomes a finite fraction of the fluid density. One direct effect of the presence of solid particles is to dampen the turbulent fluctuations, as can be inferred from the variation of von Karman's constant with sediment concentration. The constant drops from its clear fluid value of 0.4 to as low as 0.15 for a sufficiently heavy suspended load (Einstein and Chien, 1954). Einstein and Chien (1955) further advanced a modified velocity distribution equation for steady sediment-laden stream flows. On fitting with experimental data, they found it necessary to divide the flow into two zones. Close to the bed is a shallow layer of heavy-fluid zone where the sediment is heavily concentrated. Overlying this zone is a light-fluid zone where the sediment concentration is small and does not change the fluid density. The coarser the particles, the larger the fraction of the solid population residing in the heavy-fluid zone. Therefore in a concentrated sediment-laden flow the heavy-fluid zone, because of its much larger solid-water ratio, will be responsible for transportation of most of the sorbed chemicals.

The intention of the present work is to develop analytically a transport model for contaminated particles, which may originate from human dumping, in such a heavily loaded boundary layer under the pure action of progressive waves. The theory of Mei and Chian (1994) is extended to a two-layer wave boundary layer, based on the two-fluid-zone structure of Einstein and Chien (1955). We study a simplified problem in order to capture the essence of the physics. One of the idealizations is to model the abrupt yet continuous stratification by two distinct layers of homogeneous fluids; the upper fluid is clear water while the lower fluid is a denser liquid suspension. By this construction, a sharp interface between the two fluids is assumed to persist. This will not be too strong an assumption in practice as long as the stratification is stable, as should be in the present case. If the bottom layer thickness is comparable to the Monin-Obukhov length, the layer will be well mixed because the turbulence will prevail over buoyancy forces within a layer of this scale. Consequently vertical diffusion of mass may occur within the layer, but diffusion across the density interface, which forms at the top of the stable layer, is strongly inhibited. Momentum transport, on the other hand, can be transmitted across the density interface by internal waves. It has been reported that some high-density fluid mud suspensions appear to be stable, even under large shearing stresses (McCave, 1976).

2  FORMULATION

A train of two-dimensional, small-amplitude progressive waves is propagating on the surface of a clear water body of uniform depth h, which is overlying a thin heavy-fluid layer of thickness d above the bottom. The bottom is assumed to be flat, impermeable and rigid. The heavy-fluid, which is essentially a well-mixed suspension, will also be referred to as the ‘mud’ for simplicity. The flows are turbulent. We assume that near the bottom the eddy viscosities and diffusivities are constants in order to facilitate analytical development. Also, a sharp interface between the fluids is assumed, as discussed earlier.

In an (x, y) coordinate system where x is the still water level and y is pointing vertically upward, the surface profile for progressive waves propagating in the positive x-direction is given by where a is the wave amplitude, i is the imaginary unit, t is the time, k is the wavenumber and s is the wave angular frequency. Without loss of generality, a and s are assumed to be real. The flow is essentially inviscid except very near the boundaries. At the leading order, k is also real and governed by the dispersion relation  where g is the acceleration due to gravity. The near-bottom velocity given by the inviscid flow theory is  where  The vertical extents of the bottom turbulent diffusion of momentum and mass are scaled by the corresponding Stokes boundary layer thicknesses:  and , where  and are respectively the eddy diffusivity of momentum and mass. Typically the Schmidt number Sc is of order unity (supported by Reynolds analogy), and therefore the two Stokes boundary layer thicknesses are in general comparable to each other. We further assume that they are also of the same order of magnitude as the undisturbed mud depth d and the wave amplitude a, which are all much smaller than the wavelength 2pk and the water depth h. The small wave steepness  will be used as an ordering parameter. To describe flow and transport near the bottom, it is more convenient to use a local vertical coordinate , which points upward from the bottom of the mud layer. The mud-water interface is given by  where

is the interface displacement with a complex amplitude b, whose modulus is an order of magnitude smaller than a.

Let C be the concentration of a chemical in the mud, and  be the horizontal and vertical components of the mud fluid velocity. Then the mass transport of the chemical is governed by

where the eddy diffusivity of mass is assumed to be the same in horizontal and vertical directions. Since mass exchanges with the bed and the overlying clear water are ignored, the normal flux at these interfaces must be zero:

The kinematic condition at the mud-water interface is

The small ordering parameter  has been inserted in the above equations to indicate the relative order of magnitude of the associated term, as previously shown by Mei and Chian (1994). These orders are based on the following scalings:           and the Peclet number Pe  In the absence of steady currents at the leading order, the primary time scale  over which wave-induced transport becomes effective, is that due to spreading by dispersion over one horizontal length scale. This time scale is two orders of magnitude longer than a wave period since  Let us now expand the variables and the time derivative in order to obtain perturbation equations:

The leading order velocity components can be further expanded (Ng, 2000):

where and are related to each other by the continuity equation ikU_m=-dV_m/dn. A relation for the amplitude of the interface displacement is also found by substituting and into :

3  EFFECTIVE TRANSPORT EQUATIONS

The problem at O(1) is

The leading order concentration  is taken to represent the smooth component whose variations with the short time t die out in the long term (Mei and Chian, 1994; Mei et al., 1996). Hence, it is clear that on ignoring the transience  is independent of n as well, or

On simplifying using and substituting , the  problem becomes

which is subject to the boundary conditions  on  By linearity of  , the following form for  is suggested:

where the function  is governed by

In a conservative form, the perturbation equation of  reads

while the boundary conditions are

Taking depth-average followed by time-average of while using the boundary conditions yields

where the angle-brackets and the overbar denote respectively the depth-average and the time-average over a wave period T. On substituting , and , the second and third terms on the left-hand side of can be developed as follows:

where the asterisk denotes the complex conjugate, and

where and have been used. Putting these back into , we finally have the effective transport equation:

where  in which  is the depth-averaged Eulerian streaming velocity,

is an additional component for the advection velocity, and

is the dispersion coefficient. Equation governs the transport of a chemical species in the heavy-fluid layer, in which the wave action alone induces both advection and dispersion, which become effective over a long time scale . Some further analytical relationships can be developed as follows.

    We first show that the additional advection velocity component  is actually equal to the depth-averaged Stokes drift. For the present two-dimensional problem, Stokes drift (Longuet-Higgins, 1953) is given by:

whose depth average is clearly equal to :

Hence in the advection velocity, composed of Eulerian streaming velocity and Stokes drift, amounts to the depth-averaged Lagrangian drift or mass transport velocity of individual fluid particles.

    We next show that D is always positive, which is a necessary condition for a physically admissible diffusion coefficient. On substituting the equation in for  into , we may obtain in general

where integration by parts and the boundary conditions in have been used.

4  PHYSICAL DISCUSSIONS

Explicit expressions for the effective advection velocity , and the dispersion coefficient D have been deduced by Ng (2000b), based on the two-layer Stokes boundary layer theory of Ng (2000a). These expressions are functions of the wave frequency , near-bottom velocity  (which depends on the wave amplitude a, wavenumber k and water depth h), and the following dimensionless ratios: , , , and . Clearly, a heavy-fluid suspension is denser than water, so the density ratio must be less than unity: . Also,  because an eddy viscosity for wall turbulence typically increases with the distance from the wall, and also the mixing length for momentum in clear water should be greater than that in a particle-laden liquid. In general, the parameter  controls the relative rates of momentum diffusion across the mud and the water boundary layer. The ratio , or the square root of the Schmidt number, is equal to unity by Reynolds analogy. However this number can be different from unity since the mixing length for mass, because of inertia, is somewhat shorter than that for momentum, but, on the other hand, solid particles may be thrown farther to the outside of eddies by centrifugal forces (Vanoni, 1975).

The variations of the transport coefficients, in dimensionless form, with the independent parameters are examined in Figures 1-3. Non-dimensional coefficients are defined as follows:

Figure 1 shows  (solid lines) and  (dashed lines) as functions of for various  and . The value of Stokes drift can be obtained from the difference of the corresponding pair of curves. Clearly the advection velocity components increase with  and , or the advection rate is higher for a thicker layer of lighter mud. This is reasonable because for such a mud layer more fluid elements can be induced into more appreciable motion by the waves. The motion of denser mud tends to be more sluggish under wave action (Ng, 2000a). An increase in will however lower the velocities, especially for smaller and . This is so because a larger  for a fixed  means a smaller  or a smaller rate of momentum diffusion across the water boundary layer.

The effect of  on  is shown in Figure 2, where  are assumed. The dispersion coefficient reaches a maximum at a value of  which is in the order of unity, the specific values being dependent on the mud layer thickness . For instance,  is the maximum at  when , but at  when . The assumption of Reynolds analogy or  will in general lead to a dispersion coefficient that is closer to the peak value for a thicker mud layer.

Figure 3 further shows that  increases with until a maximum is reached, the values of the maximum  and the corresponding  being functions of the parameters ,  and . For the cases shown in the plots, the maximum  occurs when  is in the range 1-2.5. Also a lower  or a denser mud layer will correspond to a smaller dispersion coefficient, the effect being more prominent for a larger . Increasing the value of  will cause the peak  to occur at a smaller . In general, one may expect that the dispersion coefficient is near its peak value when the mud depth is one to two times the Stokes boundary layer thickness. The dispersion coefficient will drop significantly as the mud layer becomes too thin or too thick as compared to the Stokes boundary layer thickness.

5  CONCLUDING REMARKS

In this work, we have presented an analytical model for transport of a chemical species sorbed onto suspended sediments in a heavily-loaded benthic boundary layer under the pure action of surface progressive waves. The key assumptions are: (i) the stratified boundary layer is modeled by a two-layer boundary layer with a sharp interface; (ii) the heavy-fluid layer has a thickness comparable to the Monin-Obukhov length, the Stokes boundary layer thicknesses and the wave amplitude; (iii) fluid properties including eddy viscosity and diffusivity are constant within a fluid layer; and (iv) bedform and exchange with the bed are ignored. Following an asymptotic method of averaging which is based on the homogenization technique, we have deduced an effective transport equation, as formally given by - . The effective advection velocity is found to be equal to the depth-average of the mass transport velocity, comprising Eulerian streaming velocity and Stokes drift. Also, the dispersion coefficient can be formally shown to be positive definite. Explicit expressions for these transport coefficients are expressible as functions of characteristics of the waves and the boundary layer. We have also shown that the benthic advection and dispersion rates due to typical wind-waves alone can be as significant as those due to tidal currents. The dispersion coefficient is found to reach a maximum when the depth of the heavy-fluid is slightly thicker than the Stokes boundary layer thickness, and when the Schmidt number is slightly larger than unity.

Despite the simplifying assumptions, the present theory provides one with an analytical tool to estimate the effects of pure waves on the transport of a benthic species. Of course, other factors that have not been taken into consideration may be just as influential as the waves. A more complete model may have the individual components due to various factors lumped into the advection and dispersion coefficients. Our theory can offer a modular component due to pure waves. Future efforts to advance a more comprehensive transport model are deemed worthwhile.

Acknowledgements

The work described in this paper was supported by two grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Projects HKU 7117/99E and NSFC/HKU 8).

References

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