Xiaowei WEI , Yiming ZHANG , Changming DONG ,2,**, Meibing JIN ,2, Changshui XIA

1 School of Marine Sciences, Nanjing University of Information Science and Technology, Nanjing 210044, China

2 Southern Laboratory of Ocean Science and Engineering, Zhuhai 519000, China

3 First Institute of Oceanography, Ministry of Natural Resources, Qingdao 266101, China

Abstract Velocity vertical prof iles in the bottom boundary layer are important to understand the oceanic circulation. The logarithmic vertical prof ile, u= A ln z+ B, is the universal prof ile for the horizontal velocity in the boundary layer, in which two coeffi cients ( A and B) need to be determined. The two coeffi cients are the functions of the friction velocity ( u *) and the roughness length ( z 0), and they are calculated using u * and z 0. However, the measurement of u * and z 0 is a challenge. In the present study, an approach is developed to estimate the two coeffi cients ( A and B) by using a series of f lume laboratory experiments with f lat boundary and regularly distributed cylinders as the rough boundaries. An acoustic doppler velocimeter (ADV) is used to measure the velocity vertical prof iles of the steady f low. Using the measured velocity data, the regressed logarithmic prof iles are obtained. Based on the series of the A and B values, the mathematical formula for A and B are statistically established as the function of the cylinder height, inf low velocity, and the water depth,which avoids the measurement of the friction velocity and the roughness length.

Keyword: velocity vertical logarithmic distribution; bottom boundary layer; the friction velocity

1 INTRODUCTION

The bottom boundary layer (BBL) is a f low region in which the eff ects of friction cannot be ignored. The layer is several millimeters thick when only the molecular viscous force is considered. In a highly turbulent ocean where the turbulent viscous force is much larger than the molecular one, the thickness of the BBL can reach tens to hundreds of meters. The submarine boundary layer, which is a sink for ocean kinetic energy, is a key area for ocean dynamics studies. Investigating the distribution characteristics of the velocity in the boundary layer is of theoretical signif icance in understanding physical phenomena and mechanisms in the ocean.

It is widely accepted that the velocity prof ile near bottom boundary satisf ies the classical law of logarithmic velocity distribution (Keulegan, 1938),which is derived from the Prandtl’s mixed-length theory and it has been conf irmed in laboratory experiments and f ield studies (O’Donoghue et al.,2010; Feng et al., 2014; Valipour et al., 2015). The law of logarithmic velocity distribution is applicable for diff erent bottom boundaries in a variety of oceanic areas: the f lows above the crown of eelgrass in a bay(Lacy, 2011), the coral reef geometry (Rosman and Hench, 2011), the Gulf of California (Alvarez, 2010),and the shallow area in the East China Sea (Lozovatsky et al., 2012).

The formula for the logarithmic velocity distribution can be expressed asu=Alnz+B, which carries two coeffi cients that need to be determined in diff erent hydrological conditions. In a turbulent boundary layer, Musker (1979) uses an interpolation to combine eddy viscosity with the velocity logarithmic distribution in order to derive an explicit expression for the velocity distribution on smooth boundaries. Chen and Chiew (2004) derive a modif ied logarithmic velocity distribution formula that is applied to a boundary with seepage, is also called a boundary of suction, and their experimental data prove the reliability of their formula. By using the logarithmic law, the wake law, and the cubic correction, Guo et al. (2005) modify the logarithmic velocity distribution formula and improve results as compared to experimental data. Lozovatsky et al.(2015) discuss several diff erent logarithmic distribution models, and then f ind out that a slightly modif ied log-layer model is suitable for a tidal BBL with weak stratif ication.

The friction velocity and the roughness length are two important hydrodynamic parameters in the viscous layer near the bottom boundary, and closely related to the bottom bed roughness. Scholars have obtained several commonly used estimation methods through theoretical analysis and laboratory experiments on these two hydrodynamic parameters(Kim et al., 2000; Inoue et al., 2011; Wang et al.,2019), including Log method, turbulent kinetic energy method, and inertial dissipation method. In addition to these widely used methods, there are other estimation methods that have also been used in applications (Hao et al., 2007; Mrokowska et al.,2015). The direct estimation of the friction velocity requires high-precision measuring instruments(Leonardi et al., 2005) or has special requirements for the arrangement of the instruments (Kim et al., 2000),and there is no universally accepted empirical formula for solving roughness length directly. At the same time, since the mechanism of turbulence near the bottom boundary layer has no clear determinism,most methods are empirical. To determine the logarithmic vertical prof ile of the horizontal velocity,both the friction velocity and the roughness length need to be measured or calculated, which carries uncertainties or errors. A previous study has estimated the coeffi cientBthrough the dimensional similarity analysis method based on the measured data (So et al., 1994).

Fig.1 The wind-wave-f low f lume in the School of Marine Sciences, Nanjing University of Information Science and Technology

Based on the above discussions, the logarithmic velocity distribution needs to be determined for specif ic bottom boundary. In the present study, we conduct a series of laboratory experiments to obtain an empirical formula to determine the coeffi cients in the logarithmic velocity distribution for cylinders uniformly distributed bottom boundaries. The paper is organized as follows: Section 2 introduces the laboratory experiments and the logarithmic formula of the velocity vertical distribution based on Prandtl’s mixed-length theory; Section 3 presents the experimental results; Section 4 introduces the empirical model of two coeffi cients in the logarithmic distribution formula; and Section 5 is the conclusion of this study.

2 LABORATORY EXPERIMENT AND METHOD

2.1 Laboratory experiment

Some natural phenomena that are diffi cult to observe or cannot be directly observed can be simulated through laboratory experiments. In this paper, the velocity vertical distribution on f lat and rough bottom boundary are obtained in experiments and analyzed systematically.

2.1.1 Experimental f lume

Fig.2 Simplif ied diagram of the instruments

Fig.3 Schematic diagram of the materials used for a rough bottom bed

The experiments in this study are conducted in the wind-wave-f low f lume at School of Marine Sciences,Nanjing University of Information Science and Technology (Fig.1). The total length of the f lume is 38.4 m, the inner width is 1.4 m and the total height is 2.1 m. The base of the f lume is a brick-concrete structure and the wall of the tank is made up of glass.One end is a push plate wave maker and the other end is a fan. The bottom of the f lume is equipped with a two-way pump connected by pipes, thus allowing a f low circulation in the f lume. The velocity is adjusted by controlling the outf low current of the two-way pump. Vertical measurements of the velocity are taken 20 m away from the pump outlet to ensure a steady velocity (velocity of steady f low that does not change with time). The measurement point is mounted on a transverse bar and placed exactly in the middle between the two walls of the tank in order to reduce the inf luence of friction from the sidewalls on the velocity measurements.

2.1.2 Velocity measurement instrument

The acoustic doppler velocimeter (ADV for short,which is the Vectrino prof iler manufactured by Nortek, Fig.2), is used to measure the vertical distribution of the velocity at a f ixed position. Based on the acoustic doppler eff ect, the velocimeter emits sound pulses from the ultrasonic transmitting sensor and then calculates the instantaneous threedimensional velocity by detecting frequency or phase shifts of the coherent acoustic pulse from the three ultrasonic receiving sensors. Water molecules themselves do not ref lect sound waves. ADV uses the ref lection of sound waves by suspended particles in the water column to calculate the statistical moving speed of water parcels. Compared with traditional f low velocity detectors, ADV has the advantages of not disturbing the f low f ield, high accuracy, and short detection time (Bai et al., 2016). The ultrasonic transmitting sensor in the ADV is placed in such a way that the water sample to be measured is at least 5 cm away from the sensor. The sampling frequency of ADV is between 1 and 25 Hz, the maximum measurement range of the velocity is ±4 m/s, and the measurement accuracy is ±1 mm/s of the measured value.

2.1.3 Experimental material Plastic platforms, 2-m long and 1-m wide, f itted with plastic cylinders with uniform height are used to simulate a rough bottom bed. Diff erent rough bottom boundary conditions are made by using cylinders of diff erent heights (2, 3, 4, and 5 cm) but with equal spacing between the cylinders, as shown in Fig.3a.The ADV is placed right above the center of the rough bottom bed (Fig.3b).

2.1.4 Experiment design and data processing

This study conducts a series of experiments with diff erent water depths (25, 30, and 35 cm), bed roughness (f lat and rough with cylinders with heights of 2, 3, 4, and 5 cm) and steady velocities (pumping electronic currents set at 5, 6, 7, and 8 A, regarded as case 1, 2, 3, and 4). These experiments assume that the inf low is a steady f low, so in a f ixed point, the velocity inxdirection of vertical prof ile can be measured at diff erent times. To better resolve the velocity structure near the bottom, more measurements are conducted within the distance of 20 cm from the f lume bottom than in the area beyond the distance.The heighthrof the cylinders is used to characterize the roughness of the bottom bed,ūis the steady velocity andDis the water depth.

Fig.4 Measurements from experiments over the f lat bottom bed

As mentioned above, the ADV can measure four kinds of data. These data are recorded in thex-,y-,andz-directions. The smaller the f luctuation amplitude of the measurement, the more stable the velocity. The larger the signal-to-noise ratio and the smaller the noise in the velocity data, the better the quality of the velocity data. The higher the correlation coeffi cient and the smaller the error, the closer the data are to the real value. Measurements with a correlation coeffi cient greater than 0.7 and a signal-to-noise ratio greater than 15 are considered reliable. We control the quality of the experimental data and screen out the velocity data that meet the above conditions. A small number of spikes recorded in the data from unknown sources,possibly due to instrumentation, environmental or other factors, have been deleted.

The bottom bed platform has a thickness of about 5 cm, which is not included in the water depth. Thez=0 is set at 0.05 m over the bottom bed. Therefore, in experiments with water depths of 25, 30, and 35 cm,vertical prof iles extending up to about 20, 25, and 30 cm, respectively, are obtained.

2.2 Method

Complex turbulent motion can be decomposed into two parts: average velocity with regularity and deviation velocity. An averaging algorithm is used to transform the Navier-Stokes equations into Reynolds equations. The deviation term (Reynolds stress)caused by turbulence then appears to the Reynolds average momentum equations. In order to establish the relationship between the deviation and average values, Prandtl’s mixed-length theory analogizes turbulent motion with molecular motion, such that the f low can be regarded as composed of many waterparcels. After moving over a distance (a mixed length), the moving water parcels mixes with other parcels, exchanging energy and momentum.

Table 1 Depth-average velocities (m/s) with diff erent water depths and pumping electronic currents

According to Prandtl’s mixed-length theory and hypothesis (Umeyama and Gerritsen, 1992), the logarithmic velocity distribution equation can be derived from the eddy shear stress equation:

ParametersAandBare obtained by f itting a series of experimental results.

3 RESULT

3.1 Velocity vertical prof iles over a f lat bottom bed

Figure 4 shows the velocity vertical prof iles over the f lat bottom bed. It is observed that by applying diff erent cases, diff erent steady velocity conditions can be created. Moreover, by applying the same case but changing the water depth, the steady velocity also changes, with shallower water depth giving faster velocity. The steady velocity calculated by the data of velocity above 20 cm in the f lat experiments are summarized in Table 1. There is a depth range above the f lat bottom bed where f low velocity gradually increases as depth increases.

Fig.5 Experiments over the rough bottom boundary with 4-cm high cylinders

Fig.6 Experiments for water depth of 35 cm, on the f lat and the rough bottom bed with 2- and 4-cm high cylinders

Ideally, the f lat bottom should have no friction eff ects such that there is no transition region where the velocity changes with increasing depth. However,the ideal smooth bed conditions cannot be achieved in these actual experiments, as the designed f lat bottom bed is not perfectly f lat and still has a small level of roughness. Therefore, in the f lat bed experiments, a change in velocity in the transition zone can be observed.

3.2 Velocity vertical prof iles over rough bottom beds

Rough bottom bed experiments with diff erent cylinders heights (heights of 2, 3, 4, and 5 cm), water depths (25, 30, and 35 cm) and steady velocities(cases 1, 2, 3, and 4) are carried out. Since the velocity vertical prof iles obtained under each condition are similar (not shown), only results from experiments with 2- and 4-cm roughness height are presented(Figs.5-6).Both the steady velocity and the cylinder height aff ect velocity vertical distribution. From Figs.5-6, it is seen that the velocity near the bottom boundary is much smaller than that in the f lat bottom experiments.A zone where the velocity gradually increases with increasing depth also exists in the rough bottom experiments. A major diff erence is that the f low f ield with a distinct gradient can be observed in a rough bottom bed as opposed to a f lat bottom bed. When other environmental conditions are given, the increase in the cylinder height or the steady velocity can make the velocity prof ile steeper.

3.3 The coeffi cient of determination R 2

3.4 The friction velocity and the skin-friction coeffi cient

Fig.7 The coeffi cient of determinations R 2 of logarithmic f itting curves for 48 experiments with diff erent cylinder heights of 2, 3, 4, and 5 cm

The friction velocity is calculated using Eq.3.Figure 8 shows the relationship between the steady velocity and the friction velocity. The friction velocity is positively correlated with the steady velocity, that is, the greater the steady velocity, the greater the friction velocity. Meanwhile, it can be observed that the rougher the bottom boundary, the greater the frictional velocity, which is inversely related to the water depth.

4 DISCUSSION

According to Eq.2 and Section 3, the coeffi cient values of the logarithmic f itting formula of each experiment can be calculated and summarized. Figure 8 (scatter diagrams) shows the relationship between the coeffi cients (AandB) and the measured steady velocityū(cm/s), the cylinder heighthr(cm) over the bottom boundary and the water depthD(cm),respectively. It can be seen intuitively that the coeffi cients have an obvious linear relationship withū, but the relationships withhrandzcannot be simply expressed in a linear relationship. This study adopts the method of the curve estimation and applies several relationships (including linear relationship,logarithmic relationship, quadratic polynomial, cubic polynomial, S-shaped curve, and exponential relationship) to describe the f itting of the coeffi cients(AandB) of each group toū,hr, andD. Considering the pros and cons, it is found that the S-shaped curve has the best f itting in describingAandhr,Bandhr,while the logarithmic relationship has the best f itting in describingAandD,BandD. The average correlation coeffi cient is above 0.9.

Fig.8 Diagram of the relationship between the steady velocity and the friction velocity in the experiments with diff erent cylinder heights of 2, 3, 4, and 5 cm

Fig.9 Scatters of the skin-friction coeffi cient with diff erent cylinder heights of 2, 3, 4, and 5 cm, water depths of 25, 30, and 35 cm, the electronic currents cases of 1,2, 3, and 4

Based on the results of the curve estimation, this study constructs a multiple nonlinear regression model to describe the relationship between the coeffi cientsA,B, andū,hr,D. The model is as follows:

wherea,b, andcare f itting coeffi cients. According tothe form,AandBare positively correlated withūandhr. The largerūcauses the largerAandB, resulting in the increase of the velocity in the whole vertical prof iles. Meanwhile, the form ref lects that the increase inhrresults in stronger shear, whileAandBare inversely related withD. In addition, Eq.5 is based on the data withhr≥ 2 cm.

Table 2 Multiple nonlinear regression model of coeffi cients A and B

It can be seen from Table 2 that the model of coeffi cientsAandBhave a good f itting with the experimental data. The f itting between the predicted value of the model and the measured value can be expressed byR2and adjustedR2. TheR2of coeffi cients

AandBare 0.978 9 and 0.973 7, respectively, and adjustedR2are 0.978 0 and 0.972 5, respectively. The root mean squared error (RMSE) is a measure of the diff erence between the predicted value and the observed value. The value of RMSE is small according to Table 2, indicating that the prediction is accurate.

Figure 10 shows the comparison between the predicted values of the multiple nonlinear regression model and the experimental values. From Table 2 and Fig.10, Eq.5 can well describe the relationships betweenA,B, and the three environmental parameters and f it the measured values well. Among them,Ahas obvious correlation with the three environmental variables, that it increases with the increasing ofūandhr, and decreases with that ofD. Meanwhile, the increase ofAwithūis faster than that ofhr. The coeffi cientBand the three environmental parameters have the same correlation as that ofA, but the change is small with the increase ofhr.

5 CONCLUSION

In the present study, laboratory experiments are carried out to measure the velocity vertical prof iles over f lat and rough bottom boundaries, the distribution characteristics of velocity vertical prof iles in the bottom boundary layer are discussed, the applicability of the law of logarithmic velocity distribution in the bottom boundary layer is verif ied, and two multiple nonlinear regression models are developed for obtaining the coeffi cients of the logarithmic velocity distribution formula.It can be conf irmed by experiments that the velocity vertical prof iles over the rough bottom boundaries agree with the law of logarithmic velocity distribution.The two coeffi cientsAandBin the logarithmic velocity distribution formula follow a multi-element nonlinear relationship with the steady velocity, the cylinder height, and water depth of the rough bottom boundary. The analysis of this multivariate nonlinear regression model shows that these two coeffi cients have a linear positive correlation with the steady velocity, a positive correlation with the cylinder height in the form of S-shaped curve, and a negative correlation with the water depth in a logarithmic form. The models f it the coeffi cients that calculated by the measured data well, withR2around 0.97.

Fig.10 The comparison of coeffi cient A and B between the model predicted values (represented as “ypred” in f igure, plotted as solid lines) (a-b (c-d/e-f)) and the measured values (represented as “y” in f igure, plotted as scatters) with ū ( h r/ D)

6 DATA AVAILABILITY STATEMENT

The data that support the f indings of this study are available from the corresponding author upon request.

7 ACKNOWLEDGMENT

The measured dataset used in this study is produced by the Oceanic Modeling and Observation Laboratory of Nanjing University of information Science and Technology.