Institute of Oceanology, Chinese Academy of Sciences
Article Information
 WEI Xiaowei, ZHANG Yiming, DONG Changming, JIN Meibing, XIA Changshui
 An approach to determine coefficients of logarithmic velocity vertical profile in the bottom boundary layer
 Journal of Oceanology and Limnology, 39(6): 21442152
 http://dx.doi.org/10.1007/s003430210373x
Article History
 Received Sep. 29, 2020
 accepted in principle Nov. 11, 2020
 accepted for publication Dec. 30, 2020
2 Southern Laboratory of Ocean Science and Engineering, Zhuhai 519000, China;
3 First Institute of Oceanography, Ministry of Natural Resources, Qingdao 266101, China
The bottom boundary layer (BBL) is a flow region in which the effects 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 significance in understanding physical phenomena and mechanisms in the ocean.
It is widely accepted that the velocity profile near bottom boundary satisfies the classical law of logarithmic velocity distribution (Keulegan, 1938), which is derived from the Prandtl's mixedlength theory and it has been confirmed in laboratory experiments and field studies (O'Donoghue et al., 2010; Feng et al., 2014; Valipour et al., 2015). The law of logarithmic velocity distribution is applicable for different bottom boundaries in a variety of oceanic areas: the flows 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 as u=Alnz+B, which carries two coefficients that need to be determined in different 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 modified 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 different logarithmic distribution models, and then find out that a slightly modified loglayer model is suitable for a tidal BBL with weak stratification.
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 highprecision 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 profile 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 coefficient B through the dimensional similarity analysis method based on the measured data (So et al., 1994).
Based on the above discussions, the logarithmic velocity distribution needs to be determined for specific bottom boundary. In the present study, we conduct a series of laboratory experiments to obtain an empirical formula to determine the coefficients 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 mixedlength theory; Section 3 presents the experimental results; Section 4 introduces the empirical model of two coefficients in the logarithmic distribution formula; and Section 5 is the conclusion of this study.
2 LABORATORY EXPERIMENT AND METHOD 2.1 Laboratory experimentSome natural phenomena that are difficult to observe or cannot be directly observed can be simulated through laboratory experiments. In this paper, the velocity vertical distribution on flat and rough bottom boundary are obtained in experiments and analyzed systematically.
2.1.1 Experimental flumeThe experiments in this study are conducted in the windwaveflow flume at School of Marine Sciences, Nanjing University of Information Science and Technology (Fig. 1). The total length of the flume is 38.4 m, the inner width is 1.4 m and the total height is 2.1 m. The base of the flume is a brickconcrete 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 flume is equipped with a twoway pump connected by pipes, thus allowing a flow circulation in the flume. The velocity is adjusted by controlling the outflow current of the twoway pump. Vertical measurements of the velocity are taken 20 m away from the pump outlet to ensure a steady velocity (velocity of steady flow 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 influence of friction from the sidewalls on the velocity measurements.
2.1.2 Velocity measurement instrumentThe acoustic doppler velocimeter (ADV for short, which is the Vectrino profiler manufactured by Nortek, Fig. 2), is used to measure the vertical distribution of the velocity at a fixed position. Based on the acoustic doppler effect, 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 reflect sound waves. ADV uses the reflection of sound waves by suspended particles in the water column to calculate the statistical moving speed of water parcels. Compared with traditional flow velocity detectors, ADV has the advantages of not disturbing the flow field, 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 materialPlastic platforms, 2m long and 1m wide, fitted with plastic cylinders with uniform height are used to simulate a rough bottom bed. Different rough bottom boundary conditions are made by using cylinders of different 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 processingThis study conducts a series of experiments with different water depths (25, 30, and 35 cm), bed roughness (flat 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 inflow is a steady flow, so in a fixed point, the velocity in x direction of vertical profile can be measured at different times. To better resolve the velocity structure near the bottom, more measurements are conducted within the distance of 20 cm from the flume bottom than in the area beyond the distance. The height h_{r} of the cylinders is used to characterize the roughness of the bottom bed, ū is the steady velocity and D is the water depth.
As mentioned above, the ADV can measure four kinds of data. These data are recorded in the x, y, and zdirections. The smaller the fluctuation amplitude of the measurement, the more stable the velocity. The larger the signaltonoise ratio and the smaller the noise in the velocity data, the better the quality of the velocity data. The higher the correlation coefficient and the smaller the error, the closer the data are to the real value. Measurements with a correlation coefficient greater than 0.7 and a signaltonoise 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. The z=0 is set at 0.05 m over the bottom bed. Therefore, in experiments with water depths of 25, 30, and 35 cm, vertical profiles extending up to about 20, 25, and 30 cm, respectively, are obtained.
2.2 MethodComplex turbulent motion can be decomposed into two parts: average velocity with regularity and deviation velocity. An averaging algorithm is used to transform the NavierStokes 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 mixedlength theory analogizes turbulent motion with molecular motion, such that the flow can be regarded as composed of many water parcels. After moving over a distance (a mixed length), the moving water parcels mixes with other parcels, exchanging energy and momentum.
According to Prandtl's mixedlength theory and hypothesis (Umeyama and Gerritsen, 1992), the logarithmic velocity distribution equation can be derived from the eddy shear stress equation:
where u is the mean velocity at depth z from the bottom boundary, u_{*} is the friction velocity which is equal to
where,
Parameters A and B are obtained by fitting a series of experimental results.
3 RESULT 3.1 Velocity vertical profiles over a flat bottom bedFigure 4 shows the velocity vertical profiles over the flat bottom bed. It is observed that by applying different cases, different 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 flat experiments are summarized in Table 1. There is a depth range above the flat bottom bed where flow velocity gradually increases as depth increases.
Ideally, the flat bottom should have no friction effects 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 flat bottom bed is not perfectly flat and still has a small level of roughness. Therefore, in the flat bed experiments, a change in velocity in the transition zone can be observed.
3.2 Velocity vertical profiles over rough bottom bedsRough bottom bed experiments with different 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 profiles obtained under each condition are similar (not shown), only results from experiments with 2 and 4cm roughness height are presented (Figs. 5–6).
Both the steady velocity and the cylinder height affect velocity vertical distribution. From Figs. 5–6, it is seen that the velocity near the bottom boundary is much smaller than that in the flat bottom experiments. A zone where the velocity gradually increases with increasing depth also exists in the rough bottom experiments. A major difference is that the flow field with a distinct gradient can be observed in a rough bottom bed as opposed to a flat bottom bed. When other environmental conditions are given, the increase in the cylinder height or the steady velocity can make the velocity profile steeper.
3.3 The coefficient of determination R^{2}Statistically, the index for testing regression fitting effect is the coefficient of determination R^{2}. The R^{2} can be calculated by
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.
According to Section 2.2, if the shear stress τ_{b} on the bottom bed is determined by the roughness and equal to constant, it can be calculated by
According to Eq.2 and Section 3, the coefficient values of the logarithmic fitting formula of each experiment can be calculated and summarized. Figure 8 (scatter diagrams) shows the relationship between the coefficients (A and B) and the measured steady velocity ū (cm/s), the cylinder height h_{r} (cm) over the bottom boundary and the water depth D (cm), respectively. It can be seen intuitively that the coefficients have an obvious linear relationship with ū, but the relationships with h_{r} and z cannot 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, Sshaped curve, and exponential relationship) to describe the fitting of the coefficients (A and B) of each group to ū, h_{r}, and D. Considering the pros and cons, it is found that the Sshaped curve has the best fitting in describing A and h_{r}, B and h_{r}, while the logarithmic relationship has the best fitting in describing A and D, B and D. The average correlation coefficient is above 0.9.
Based on the results of the curve estimation, this study constructs a multiple nonlinear regression model to describe the relationship between the coefficients A, B, and ū, h_{r}, D. The model is as follows:
where a, b, and c are fitting coefficients. According to the form, A and B are positively correlated with ū and h_{r}. The larger ū causes the larger A and B, resulting in the increase of the velocity in the whole vertical profiles. Meanwhile, the form reflects that the increase in h_{r} results in stronger shear, while A and B are inversely related with D. In addition, Eq.5 is based on the data with h_{r} 2 cm.
It can be seen from Table 2 that the model of coefficients A and B have a good fitting with the experimental data. The fitting between the predicted value of the model and the measured value can be expressed by R^{2} and adjusted R^{2}. The R^{2} of coefficients A and B are 0.978 9 and 0.973 7, respectively, and adjusted R^{2} are 0.978 0 and 0.972 5, respectively. The root mean squared error (RMSE) is a measure of the difference 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 between A, B, and the three environmental parameters and fit the measured values well. Among them, A has obvious correlation with the three environmental variables, that it increases with the increasing of ū and h_{r}, and decreases with that of D. Meanwhile, the increase of A with ū is faster than that of h_{r}. The coefficient B and the three environmental parameters have the same correlation as that of A, but the change is small with the increase of h_{r}.
5 CONCLUSIONIn the present study, laboratory experiments are carried out to measure the velocity vertical profiles over flat and rough bottom boundaries, the distribution characteristics of velocity vertical profiles in the bottom boundary layer are discussed, the applicability of the law of logarithmic velocity distribution in the bottom boundary layer is verified, and two multiple nonlinear regression models are developed for obtaining the coefficients of the logarithmic velocity distribution formula.
It can be confirmed by experiments that the velocity vertical profiles over the rough bottom boundaries agree with the law of logarithmic velocity distribution. The two coefficients A and B in the logarithmic velocity distribution formula follow a multielement 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 coefficients have a linear positive correlation with the steady velocity, a positive correlation with the cylinder height in the form of Sshaped curve, and a negative correlation with the water depth in a logarithmic form. The models fit the coefficients that calculated by the measured data well, with R^{2} around 0.97.
6 DATA AVAILABILITY STATEMENTThe data that support the findings of this study are available from the corresponding author upon request.
7 ACKNOWLEDGMENTThe measured dataset used in this study is produced by the Oceanic Modeling and Observation Laboratory of Nanjing University of information Science and Technology.
Alvarez L G. 2010. Bottom boundary layer properties in the Upper Gulf of California derived from velocity profiles. Ciencias Marinas, 36(3): 285299.
DOI:10.7773/cm.v36i3.1760 
Bai R N, Cao L K, Li D X. 2016. Applicability of Vectrino Profiler in measurement of open channel flows. Journal of Hydroelectric Engineering, 35(11): 3544.
(in Chinese with English abstract) 
Chen X W, Chiew Y M. 2004. Velocity distribution of turbulent openchannel flow with bed suction. Journal of Hydraulic Engineering, 130(2): 140148.
DOI:10.1061/(ASCE)07339429(2004)130:2(140) 
Feng D J, Wu L H, Shimozono T, Okayasu A. 2014. Detailed measurements of boundary layer flow under swash with highresolution PIV. Journal of Japan Society of Civil Engineers Ser B2 (Coastal Engineering), 70(2): I_741I_745.
DOI:10.2208/kaigan.70.I_741 
Guo J K, Julien P Y, Meroney R N. 2005. Modified logwake law for zeropressuregradient turbulent boundary layers. Journal of Hydraulic Research, 43(4): 421430.
DOI:10.1080/00221680509500138 
Hao J L, Song Z Y, Yan Y X, Li H Q. 2007. Study on the tidal velocity profile in the estuarine and coastal areas. Journal of Sediment Research, (4): 3441.

Inoue T, Glud R N, Stahl H, Hume A. 2011. Comparison of three different methods for assessing in situ friction velocity: A case study from Loch Etive, Scotland. Limnology and Oceanography: Methods, 9(6): 275287.
DOI:10.4319/lom.2011.9.275 
Keulegan G H. 1938. Laws of turbulent flow in open Channels. Journal of Research of the National Bureau of Standards, 21(6): 707741.
DOI:10.6028/jres.021.039 
Kim S C, Friedrichs C T, Maa J P Y, Wright L D. 2000. Estimating bottom stress in tidal boundary layer from acoustic Doppler velocimeter data. Journal of Hydraulic Engineering, 126(6): 399406.
DOI:10.1061/(ASCE)07339429(2000)126:6(399) 
Kirkgöz M S. 1989. Turbulent velocity profiles for smooth and rough open channel flow. Journal of Hydraulic Engineering, 115(11): 15431561.
DOI:10.1061/(ASCE)07339429(1989)115:11(1543) 
Lacy J R. 2011. The influence of current speed and vegetation density on flow structure in two macrotidal eelgrass canopies. Limnology and Oceanography: Fluids and Environments, 1(1): 3855.
DOI:10.1215/215736981152489 
Lozovatsky I, Jinadasa S U P, Fernando H J S, Lee J H, Chang S. 2015. The walllayer dynamics in a weakly stratified tidal bottom boundary layer. Journal of Marine Research, 73(6): 207232.
DOI:10.1357/002224015817391276 
Lozovatsky I, Liu Z Y, Fernando H, Armengol J, Roget E. 2012. Shallow water tidal currents in close proximity to the seafloor and boundaryinduced turbulence. Ocean Dynamics, 62(2): 177191.
DOI:10.1007/s1023601104953 
Mrokowska M M, Rowiński P M, Kalinowska M B. 2015. Evaluation of friction velocity in unsteady flow experiments. Journal of Hydraulic Research, 53(5): 659669.
DOI:10.1080/00221686.2015.1072853 
Musker A J. 1979. Explicit expression for the smooth wall velocity distribution in a turbulent boundary layer. AIAA Journal, 17(6): 655657.
DOI:10.2514/3.61193 
O'Donoghue T, Pokrajac D, Hondebrink L J. 2010. Laboratory and numerical study of dambreakgenerated swash on impermeable slopes. Coastal Engineering, 57(5): 513530.
DOI:10.1016/j.coastaleng.2009.12.007 
Rosman J H, Hench J L. 2011. A framework for understanding drag parameterizations for coral reefs. Journal of Geophysical Research: Oceans, 116(C8): C08025.
DOI:10.1029/2010JC006892 
Schlichting H, Gersten K. 2017. BoundaryLayer Theory. Springer, Berlin, Heidelberg. p.150155.
DOI:10.1007/9783662529195

So R M C, Zhang H S, Gatski T B, Speziale C G. 1994. Logarithmic laws from compressible turbulent boundary layers. AIAA Journal, 32(11): 21622168.
DOI:10.2514/3.12273 
Umeyama M, Gerritsen F. 1992. Velocity distribution in uniform sedimentladen flow. Journal of Hydraulic Engineering, 118(2): 229245.
DOI:10.1061/(ASCE)07339429(1992)118:2(229) 
Valipour R, Bouffard D, Boegman L. 2015. Parameterization of bottom mixed layer and logarithmic layer heights in central Lake Erie. Journal of Great Lakes Research, 41(3): 707718.
DOI:10.1016/j.jglr.2015.06.010 
Wang K J, Bai J X, Tang Z Q, Jiang N. 2019. Comparative study of turbulent boundary layer wall friction velocity measured by average velocity profile method. Journal of Experimental Mechanics, 34(2): 209216.
(in Chinese with English abstract) 
Wang X Y, Yang Q Y, Lu W Z, Wang X K. 2012. Experimental study of nearwall turbulent characteristics in an openchannel with gravel bed using an acoustic Doppler velocimeter. Experiments in Fluids, 52(1): 8594.
DOI:10.1007/s0034801112023 
Wang Y P, Gao S, Jia J J. 2000. Flow structure in the marine boundary layer and bedload transport: A review. Marine Geology & Quaternary Geology, 20(3): 101106.
(in Chinese with English abstract) 