Effects of Hydrodynamic Conditions on DO Transfer at a Rough Sediment Surface

The experiment was performed by means of a rectangular flume system in order to investigate the effects of surface roughness on diffusion of DO across the sediment-water interface. The experimental devices involved a rectangular flume (250 cm long, 12.5 cm wide, and 15 cm deep), in which flow velocity could be controlled (Fig. 1). This rectangular flume was hermetically sealed, and the balance of dissolved gases can be calculated. The sediment was placed in a cavity (100 cm long, 12.5 cm wide, and 10 cm deep) in the middle section of the flume bed (see also Inoue and Nakamura 2009).

Sediment was collected with a shovel from a eutrophic riverbed and immediately transported to the laboratory. Sediment was placed in the aforementioned cavity to the same height as the flume bed. Fresh water (saturated with DO) was poured into the flume taking care not to disturb the sediment surface. Then, water was circulated at constant flow rate controlled by a pump and valve. The water temperature was maintained at $30\pm 0.1°\text{C}$ by a temperature controller (ORION, LPA3, Suzaka City, Nagano, Japan). The experiment was started after confirming that benthic animals (river crabs and nereimorpha) were removed by means of tweezers as thoroughly as possible.

Seventeen experimental runs were completed to test three different (0 mm, 3 mm, and 5 mm) roughness conditions at six different flow velocities (Table 1) to test multiple roughness Reynolds numbers. Roughness was altered by inserting squares of 3 mm and 5-mm-high Perspex at intervals of 50 mm, fixed with narrow bars (0.8 mm in diameter) needled through both edges. Bandyopadhyay (1987) defined roughness as $k$ type (the ratio of cavity length/roughness $\text{height}>3$ ) and $d$ type (the ratio of cavity length/roughness $\text{height}<1$ ), and this experiment could be considered to be the $k$ type. As this flume was too small for turbulence to fully develop, flow velocity distribution above the sediment section was measured by a laser Doppler velocimeter (TSI, Model9710, Shoreview, Minn.), and the relationship between the cross-sectional mean flow velocity $\left(\overline{u}\right)$ and the spatially averaged shear velocity $\left(u_{\ast }\right)$ and equivalent sand roughness $\left(k_s\right)$ were obtained on the basis of a logarithmic profile of flow velocity (Grant et al. 1984).

Photosynthesis was suppressed by covering the flume with a thick black curtain. DO concentration was monitored every hour during the experiment with a DO meter (DKK-TOA Corp., DO-25A, Shinjuku-ku, Tokyo) and the probe was installed at the downstream end of the flume (Fig. 1). The DO concentration and the meter calibration was checked by Winkler method at start and finish of each experimental run. After each experimental run, air bubbles were passed through the flume water to increase DO.

The overlying water in the flume was sampled from the sampling cock without being exposed to the air at approximately one-day intervals. The DO consumption rate per unit volume of the overlying water, $r_w$ , was measured using a quasi-biological oxygen demand method, at a temperature of $30°\text{C}$ , and the effect of respiration in the water was corrected. Using the following equation, SOD was calculated using decreasing rate of DO concentration:where $V=\text{the}$ entire volume of recirculating water in the system; $C_O=\text{DO}$ concentration; $t=\text{time}$ ; $A=\text{surface}$ area of the sediment; and $r_w=\text{volume}$ specific oxygen consumption rate in the water.

The porosity and volume specific oxygen consumption rate of the sediment were measured immediately after the completion of each experiment. Known volumes of sediment layers were quarried from narrow sediment cores using end-cut syringes. Some portions of the sediment samples were dried at $60°\text{C}$ for 2 days and the porosity was estimated from the weight difference before and after the drying process. Other samples were used to measure the volume specific oxygen consumption rate of the sediment, $r_s$ , which was determined by the method of Hosoi et al. (1992). This method involved measuring the DO concentration of water clouded with sediment. Experimental conditions are listed in Table 1.

DO must traverse through two layers from the water column to the sediment, before consumption through respiration and chemical reactions in the sediment. The first layer is the DBL, located in the water immediately above the sediment surface. The second is the surface layer of the sediment, which is typically 2–3 mm deep for eutrophicated sediment. DO concentration at the sediment-water interface is affected by physical conditions above the interface and by sedimentary oxygen consumption kinetics. This was modeled by examining diffusion in each layer separately and then combining the results to calculate the diffusion rate of DO from the bulk water to the sediment.

Sample sediments had a porosity of approximately 0.76, and volume specific oxygen consumption rate, $r_s$ , of $69\mu \text{mol}{\text{cm}}^{-3}{\text{s}}^{-1}$ . $\mathsf{Sc}$ was 301 (since water temperature was maintained at $30°\text{C}$ ). These parameters are used in the following analyses and discussion. In all experimental runs, most DO in the overlying water was consumed by sedimentary oxygen demand and DO concentration in the bulk region decreased with time. SOD was calculated from the rate of DO concentration decline using Eq. (1).

Fig. 4 shows the relationship between ${\mathsf{R}}_k$ and the $\mathsf{St}$ obtained from the experiment results, $\mathsf{S}{\mathsf{t}}_{\text{ex}}$ . Although the calculated $\mathsf{S}{\mathsf{t}}_{\text{ex}}$ varies widely, $\mathsf{S}{\mathsf{t}}_{\text{ex}}$ increases with ${\mathsf{R}}_k$ under a smooth surface condition $\left({\mathsf{R}}_k<30\right)$ ; however, $\mathsf{S}{\mathsf{t}}_{\text{ex}}$ has a maximum value (two to five times larger than that on the smooth surface) in the buffer region between the smooth and complete rough surface $\left(50<{\mathsf{R}}_k<70\right)$ and declines in rough regions $\left({\mathsf{R}}_k>100\right)$ . This tendency is similar to that described in previous papers (Dade 1993; Zhao and Trass 1997). In low ${\mathsf{R}}_k$ regions, kinetic viscosity—i.e., turbulent diffusivity—in the boundary layer is relatively small, and the DO concentration gradient in the DBL is relatively gentle. In addition, exchanges of water between the bulk region and the semistagnant layer are not frequent because the shear stress acting on the water between the roughness elements is relatively small. As a result, the DO transfer rate is small in regions of low ${\mathsf{R}}_k$ . As ${\mathsf{R}}_k$ gets larger, the shear stress acting on the water between the roughness elements also gets larger. Large shear stresses make the exchanges of water between the bulk region and the semistagnant layer more frequent and DO transfer increases. In the fully rough region, the effect of roughness elements becomes relatively large. Therefore, water exchanges between the bulk region and the semistagnant layer are not so frequent and water between the roughness elements become “dead water,” although kinetic viscosity and turbulent diffusivity in the boundary layer are high. Since the dead water is resistant to the diffusion of DO, the DO transfer is suppressed in the fully rough region. This tendency is qualitatively the same as that of heat (e.g., Dipprey and Sabersky 1963) and mass (e.g., Dawson and Trass 1972) transfer across the rough wall.

In Fig. 4, $\mathsf{St}$ values from Dawson and Trass (1972) are also plotted for each experimental condition, from the equationwhere ${\mathsf{Sc}}_R=\text{Stanton}$ number for a rough surface and ${\mathsf{Sc}}_S=\text{Stanton}$ number for a smooth surface. The $\mathsf{St}$ values estimated by Eq. (27) are in the range of our experimental results, but do not fit experimental data. Bilger and Atkinson (1992) also proposed a model, but their values were around 0.0058, considerably larger than the $\mathsf{S}{\mathsf{t}}_{\text{ex}}$ observed in our experiment and are not shown in Fig. 4.

Fig. 5 shows an example of the fluctuation in DO diffusion at the sediment-water interface obtained from a nonsteady calculation over 10 s. In this example, experimental conditions of Ex#1 were used as initial parameters for nonsteady calculation. SOD achieves quasi-steady state after the occurrence of fourth vortex shedding and SOD is enhanced by up to almost twice the initial value, which corresponds with estimated value obtained by modeling excluding the nonsteady concept following the vortex shedding event. Time-averaged value under the quasi-steady state is 1.86 times initial value. The maxima develop approximately 0.6 s after the occurrence of each vortex shedding, and the monotonous increase of SOD in the initial stage is caused by an increase in the DO concentration gradient in $0<z<\beta {\delta }_d$ , due to the rapid diffusion of DO from the renewed water in $z>\beta {\delta }_d$ . After that, SOD decreased monotonously, as the DO concentration gradient in $0<z<{\delta }_d$ decreased due to the development of the DBL. The fluctuation of the DO diffusion rate is explained by variation in the vertical DO profile in the vicinity of the sediment-water interface before and after the occurrence of vortex shedding (Fig. 3).

From these calculation results, the averaged SOD values under quasi-steady state were calculated, and the relationship between the experimental and calculation results of $\mathsf{St}$ are shown in Fig. 6(A). The experimental results were well correlated with calculation results $\left(R^2=0.786\right)$ , that is, the average error was about 6.5%. The importance of the respective formulated procedures stated above will be discussed below.

In this paper, DO diffusion equation in the sediment, Eq. (15), was vertically integrated in the oxic layer assuming a steady state. First, the validity of the assumption is discussed. Using Eq. (17), the time scale of diffusion in the sediment, ${\tau }_s$ , is defined byUsing experimental values, ${\tau }_s$ was calculated to be 174–566 s. Over this time, DO concentration in the flume decreased $0.3–0.9\mu \text{mol}{\text{L}}^{-1}$ , which correspond to 0.1–0.4% of the DO concentration in the bulk region. Therefore, we concluded the time scale required for DO profile to achieve a steady state in the sediment surface was short enough to assume a steady state in vertical integration procedure of Eq. (15). This implies the assumption of a steady state in the sediment model is valid.

Fig. 6(B) depicts the relationship between the experimental and calculation results excluding the nonsteady concept following a vortex shedding event. In these calculations, we assumed that $c_1=1$ (as did Dade 1993). The calculation results underestimated the experimental results by approximately 64.2% [Fig. 6(B)], and the correlation coefficient was low $\left(R^2=0.501\right)$ . The low correlation may have two causes: the quantification of flushing interval, $s$ (that is $c_1$ ) and the effect of nonsteady concept following the vortex shedding event are discussed below.

In this section, the effect of quantification of the flushing interval of dead water between roughness elements, $s$ , was considered. Corino and Brodkey (1969) investigated the fluid motions of a turbulent flow near a solid boundary, and revealed that the viscous sublayer was continuously disturbed by small-scale velocity fluctuations and periodically renewed by fluid elements that penetrated into the viscous sublayer from positions away from the wall. Danckwerts (1951) introduced term “surface renewal” to describe this phenomenon. Corino and Brodkey (1969) considered these “bursts” to be the most important feature of the wall region, and the flushing frequency is believed to be critical factor governing the mass transfer at the surface of the wall. Under rough surface conditions, vortex shedding may be an important factor instead of burst phenomenon. Therefore, in this section, the effect of quantitative estimation of $s$ and $c_1$ on the calculation of $\mathsf{St}$ is discussed.

As $s$ can be calculated by Eq. (13), DO flux, $J$ , and $\mathsf{St}$ can be estimated from Eqs. (9) - (14). Fig. 6(C) shows the relation between the experimental and calculation results when Eq. (13) is used for the quantitative estimation of $s$ , but neglecting the nonsteady concept following the vortex shedding event. Although the calculation results still underestimate experimental results by approximately 63.0%, the correlation coefficient increases, where $R^2=0.787$ . From this result, we consider that the scatter of theoretically estimated values [shown in Fig. 6(B)] is attributable to inaccurate estimation of the flushing interval of dead water between roughness elements (i.e., the occurrence interval of the vortex shedding) through the model formulation procedure.

In this formulation, the numerical constant, $c_1$ , can also be obtained as a simple function of ${\mathsf{R}}_k$ by Eq. (29).The $c_1$ values calculated for each experimental condition are listed in Table 2.

The averaged SOD values under the quasi-steady state were obtained from the same procedure as shown in Fig. 5. In this section, although the estimation of $s$ enables us to quantify the $c_1$ value, the improvement in model accuracy can be estimated by comparison of exclusion and inclusion of the adaptation of the nonsteady concept following the vortex shedding event. The relationship between experimental and calculation results for $\mathsf{St}$ is shown in Fig. 6(D). The underestimation of the calculation results shown in Fig. 6(B) is greatly improved, and the averaged error is about 12.6%. Overall underestimation of $\mathsf{St}$ can be attributed to exclusion of nonsteady variations in SOD after the occurrence of vortex shedding. However, without the quantitative formulation of $s$ and $c_1$ , the correlation coefficient is low $\left(R^2=0.433\right)$ .

The monotonous increase of SOD immediately after the occurrence of vortex shedding is caused by an increase in the DO concentration gradient in $0<z<\beta {\delta }_d$ , due to the rapid diffusion of DO from the volume of renewed water where $z>\beta {\delta }_d$ . This is a temporary phenomenon and the time scale, ${\tau }_{d1}$ , can be calculated fromwhere $\overline{D_{\text{zt}1}}=\text{vertically}$ averaged turbulent diffusivity in the range of $0<z<\beta {\delta }_d$ . After that time, the steady decrease in SOD was caused by a decreasing DO concentration gradient in $0<z<{\delta }_d$ due to the development of the DBL. This is also a temporary phenomenon, and the time scale, ${\tau }_{d2}$ , of this phenomenon can be calculated fromwhere $\overline{D_{\text{zt}2}}=\text{vertically}$ averaged turbulent diffusivity in the range of $0<z<{\delta }_d$ . The calculated ${\tau }_{d1}$ and ${\tau }_{d2}$ values for each experiment are listed in Table 2.

From the discussion, the reproducibility of the experimental results by numerical analysis is significantly improved by the quantitative formulation of the flushing frequency of the water in the cavity between the roughness elements (i.e., vortex shedding), considering nonsteady variations in the material transfer rate due to step changes in the material concentration of the overlying water. These concepts are applicable to other models describing heat and mass transfer at the solid-liquid interface.

The numerical calculations of nonsteady variations after the occurrence of vortex shedding shown in the previous sections are complicated and impractical and a simple quantification of the enhancement effect for $\mathsf{St}$ is desirable. Fig. 7 shows the relationship between $u_{\ast }$ and the enhancement factor, $F$ , for $\mathsf{St}$ due to nonsteady variations within the range of experimental conditions in the present study $\left(0.2\text{cm}{\text{s}}^{-1}<u_{\ast }<3.6\text{cm}{\text{s}}^{-1}\right)$ . The enhancement factor, $F$ , is a monotonously decreasing function of $u_{\ast }$ . The variation is aptly reproduced by a regression curve, which is formulated as a quadratic equation of $u_{\ast }$ , as follows:The enhancement factor, $F$ , does not show a clear relationship with the equivalent sand roughness, $k_s$ , and other parameters. $\mathsf{St}$ values without nonsteady calculations can be simply revised from Eq. (32), based on bed shear stress information.